Runtime Complexity (innermost) proof of /tmp/tmpmkgl7A/Ex6_15_AEL02

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 186 ms)

↳2 CdtProblem

↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 14 ms)

↳4 CdtProblem

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

↳6 CdtProblem

↳7 CdtUnreachableProof (⇔, 0 ms)

↳8 CdtProblem

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

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 109 ms)

↳12 CdtProblem

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

↳14 CdtProblem

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

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 498 ms)

↳18 CdtProblem

↳19 CdtUnreachableProof (⇔, 0 ms)

↳20 CdtProblem

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

↳22 CdtProblem

↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 254 ms)

↳24 CdtProblem

↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 400 ms)

↳26 CdtProblem

↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 336 ms)

↳28 CdtProblem

↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 363 ms)

↳30 CdtProblem

↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 319 ms)

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 278 ms)

↳34 CdtProblem

↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 248 ms)

↳36 CdtProblem

↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 255 ms)

↳38 CdtProblem

↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 217 ms)

↳40 CdtProblem

↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 219 ms)

↳42 CdtProblem

↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 218 ms)

↳44 CdtProblem

↳45 SIsEmptyProof (⇔, 0 ms)

↳46 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c4(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(s(z0))) → c12(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote(s1(z0))) → c16(S(unquote(z0)), UNQUOTE(z0))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c4(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(s(z0))) → c12(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote(s1(z0))) → c16(S(unquote(z0)), UNQUOTE(z0))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

ACTIVE, SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, PROPER, QUOTE, QUOTE1, TOP

Compound Symbols:

c, c3, c4, c5, c6, c8, c10, c12, c13, c14, c16, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c86, c87, c88, c89

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

Removed 4 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

ACTIVE, SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, PROPER, QUOTE, QUOTE1, TOP

Compound Symbols:

c, c3, c5, c6, c8, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c86, c87, c88, c89, c4, c12, c16

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

Use narrowing to replace ACTIVE(first1(s(z0), cons(z1, z2))) → c8(CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2)) by

ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(mark(z0)), cons(x1, z1))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(mark(z0), z1))
ACTIVE(first1(s(ok(z0)), cons(x1, ok(z1)))) → c8(CONS1(quote(x1), ok(first1(z0, z1))), QUOTE(x1), FIRST1(ok(z0), ok(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
ACTIVE(first1(s(x0), cons(x1, x2))) → c8

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(mark(z0)), cons(x1, z1))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(mark(z0), z1))
ACTIVE(first1(s(ok(z0)), cons(x1, ok(z1)))) → c8(CONS1(quote(x1), ok(first1(z0, z1))), QUOTE(x1), FIRST1(ok(z0), ok(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
ACTIVE(first1(s(x0), cons(x1, x2))) → c8

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(mark(z0)), cons(x1, z1))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(mark(z0), z1))
ACTIVE(first1(s(ok(z0)), cons(x1, ok(z1)))) → c8(CONS1(quote(x1), ok(first1(z0, z1))), QUOTE(x1), FIRST1(ok(z0), ok(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
ACTIVE(first1(s(x0), cons(x1, x2))) → c8

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

ACTIVE, SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, PROPER, QUOTE, QUOTE1, TOP

Compound Symbols:

c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c86, c87, c88, c89, c4, c12, c16, c8, c8

(7) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(first1(s(mark(z0)), cons(x1, z1))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(mark(z0), z1))
ACTIVE(first1(s(ok(z0)), cons(x1, ok(z1)))) → c8(CONS1(quote(x1), ok(first1(z0, z1))), QUOTE(x1), FIRST1(ok(z0), ok(z1)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
ACTIVE(first1(s(x0), cons(x1, x2))) → c8
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
ACTIVE(first1(s(x0), cons(x1, x2))) → c8

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, TOP, ACTIVE, PROPER

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c88, c89, c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c4, c12, c16, c8, c8, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85

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

Removed 1 trailing nodes:

ACTIVE(first1(s(x0), cons(x1, x2))) → c8

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, TOP, ACTIVE, PROPER

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c88, c89, c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c4, c12, c16, c8, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85

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

Use narrowing to replace TOP(mark(z0)) → c88(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)), PROPER(0))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c88(TOP(ok(nil)), PROPER(nil))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)), PROPER(nil1))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(01)) → c88(TOP(ok(01)), PROPER(01))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(x0)) → c88

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)), PROPER(0))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c88(TOP(ok(nil)), PROPER(nil))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)), PROPER(nil1))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(01)) → c88(TOP(ok(01)), PROPER(01))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(x0)) → c88

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)), PROPER(0))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c88(TOP(ok(nil)), PROPER(nil))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)), PROPER(nil1))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(01)) → c88(TOP(ok(01)), PROPER(01))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(x0)) → c88

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, TOP, ACTIVE, PROPER

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c89, c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c4, c12, c16, c8, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c88, c88

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c88

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)), PROPER(0))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c88(TOP(ok(nil)), PROPER(nil))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)), PROPER(nil1))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(01)) → c88(TOP(ok(01)), PROPER(01))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)), PROPER(0))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c88(TOP(ok(nil)), PROPER(nil))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)), PROPER(nil1))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(01)) → c88(TOP(ok(01)), PROPER(01))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, TOP, ACTIVE, PROPER

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c89, c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c4, c12, c16, c8, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c88

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

Removed 4 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, TOP, ACTIVE, PROPER

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c89, c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c4, c12, c16, c8, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c88, c88

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

Use narrowing to replace TOP(ok(z0)) → c89(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(sel(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(sel(0, cons(z0, z1)))) → c89(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(first(0, z0))) → c89(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c89(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(sel1(0, cons(z0, z1)))) → c89(TOP(mark(quote(z0))), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(first1(0, z0))) → c89(TOP(mark(nil1)), ACTIVE(first1(0, z0)))
TOP(ok(first1(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons1(quote(z1), first1(z0, z2)))), ACTIVE(first1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c89(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote1(cons(z0, z1)))) → c89(TOP(mark(cons1(quote(z0), quote1(z1)))), ACTIVE(quote1(cons(z0, z1))))
TOP(ok(quote1(nil))) → c89(TOP(mark(nil1)), ACTIVE(quote1(nil)))
TOP(ok(quote(s(z0)))) → c89(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(sel(z0, z1)))) → c89(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(quote1(first(z0, z1)))) → c89(TOP(mark(first1(z0, z1))), ACTIVE(quote1(first(z0, z1))))
TOP(ok(unquote(01))) → c89(TOP(mark(0)), ACTIVE(unquote(01)))
TOP(ok(unquote(s1(z0)))) → c89(TOP(mark(s(unquote(z0)))), ACTIVE(unquote(s1(z0))))
TOP(ok(unquote1(nil1))) → c89(TOP(mark(nil)), ACTIVE(unquote1(nil1)))
TOP(ok(unquote1(cons1(z0, z1)))) → c89(TOP(mark(fcons(unquote(z0), unquote1(z1)))), ACTIVE(unquote1(cons1(z0, z1))))
TOP(ok(fcons(z0, z1))) → c89(TOP(mark(cons(z0, z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(s(z0))) → c89(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c89(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(from(z0))) → c89(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(active(z0), z1)), ACTIVE(first1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(z0, active(z1))), ACTIVE(first1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))
TOP(ok(s1(z0))) → c89(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(unquote(z0))) → c89(TOP(unquote(active(z0))), ACTIVE(unquote(z0)))
TOP(ok(unquote1(z0))) → c89(TOP(unquote1(active(z0))), ACTIVE(unquote1(z0)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(active(z0), z1)), ACTIVE(fcons(z0, z1)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(z0, active(z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(x0)) → c89

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(sel(0, cons(z0, z1)))) → c89(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(first(0, z0))) → c89(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c89(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(sel1(0, cons(z0, z1)))) → c89(TOP(mark(quote(z0))), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(first1(0, z0))) → c89(TOP(mark(nil1)), ACTIVE(first1(0, z0)))
TOP(ok(first1(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons1(quote(z1), first1(z0, z2)))), ACTIVE(first1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c89(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote1(cons(z0, z1)))) → c89(TOP(mark(cons1(quote(z0), quote1(z1)))), ACTIVE(quote1(cons(z0, z1))))
TOP(ok(quote1(nil))) → c89(TOP(mark(nil1)), ACTIVE(quote1(nil)))
TOP(ok(quote(s(z0)))) → c89(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(sel(z0, z1)))) → c89(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(quote1(first(z0, z1)))) → c89(TOP(mark(first1(z0, z1))), ACTIVE(quote1(first(z0, z1))))
TOP(ok(unquote(01))) → c89(TOP(mark(0)), ACTIVE(unquote(01)))
TOP(ok(unquote(s1(z0)))) → c89(TOP(mark(s(unquote(z0)))), ACTIVE(unquote(s1(z0))))
TOP(ok(unquote1(nil1))) → c89(TOP(mark(nil)), ACTIVE(unquote1(nil1)))
TOP(ok(unquote1(cons1(z0, z1)))) → c89(TOP(mark(fcons(unquote(z0), unquote1(z1)))), ACTIVE(unquote1(cons1(z0, z1))))
TOP(ok(fcons(z0, z1))) → c89(TOP(mark(cons(z0, z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(s(z0))) → c89(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c89(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(from(z0))) → c89(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(active(z0), z1)), ACTIVE(first1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(z0, active(z1))), ACTIVE(first1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))
TOP(ok(s1(z0))) → c89(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(unquote(z0))) → c89(TOP(unquote(active(z0))), ACTIVE(unquote(z0)))
TOP(ok(unquote1(z0))) → c89(TOP(unquote1(active(z0))), ACTIVE(unquote1(z0)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(active(z0), z1)), ACTIVE(fcons(z0, z1)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(z0, active(z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(x0)) → c89

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(sel(0, cons(z0, z1)))) → c89(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(first(0, z0))) → c89(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c89(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(sel1(0, cons(z0, z1)))) → c89(TOP(mark(quote(z0))), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(first1(0, z0))) → c89(TOP(mark(nil1)), ACTIVE(first1(0, z0)))
TOP(ok(first1(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons1(quote(z1), first1(z0, z2)))), ACTIVE(first1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c89(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote1(cons(z0, z1)))) → c89(TOP(mark(cons1(quote(z0), quote1(z1)))), ACTIVE(quote1(cons(z0, z1))))
TOP(ok(quote1(nil))) → c89(TOP(mark(nil1)), ACTIVE(quote1(nil)))
TOP(ok(quote(s(z0)))) → c89(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(sel(z0, z1)))) → c89(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(quote1(first(z0, z1)))) → c89(TOP(mark(first1(z0, z1))), ACTIVE(quote1(first(z0, z1))))
TOP(ok(unquote(01))) → c89(TOP(mark(0)), ACTIVE(unquote(01)))
TOP(ok(unquote(s1(z0)))) → c89(TOP(mark(s(unquote(z0)))), ACTIVE(unquote(s1(z0))))
TOP(ok(unquote1(nil1))) → c89(TOP(mark(nil)), ACTIVE(unquote1(nil1)))
TOP(ok(unquote1(cons1(z0, z1)))) → c89(TOP(mark(fcons(unquote(z0), unquote1(z1)))), ACTIVE(unquote1(cons1(z0, z1))))
TOP(ok(fcons(z0, z1))) → c89(TOP(mark(cons(z0, z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(s(z0))) → c89(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c89(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(from(z0))) → c89(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(active(z0), z1)), ACTIVE(first1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(z0, active(z1))), ACTIVE(first1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))
TOP(ok(s1(z0))) → c89(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(unquote(z0))) → c89(TOP(unquote(active(z0))), ACTIVE(unquote(z0)))
TOP(ok(unquote1(z0))) → c89(TOP(unquote1(active(z0))), ACTIVE(unquote1(z0)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(active(z0), z1)), ACTIVE(fcons(z0, z1)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(z0, active(z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(x0)) → c89

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, ACTIVE, PROPER, TOP

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c, c3, c5, c6, c10, c13, c14, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c4, c12, c16, c8, c68, c69, c70, c72, c74, c75, c76, c77, c79, c81, c82, c83, c84, c85, c88, c88, c89, c89

(19) 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))) → c(SEL(z0, z2))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(QUOTE(z0))
ACTIVE(quote1(cons(z0, z1))) → c10(CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote(sel(z0, z1))) → c13(SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(FIRST1(z0, z1))
ACTIVE(unquote1(cons1(z0, z1))) → c18(FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(CONS(z0, z1))
ACTIVE(sel(z0, z1)) → c20(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c21(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c22(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c23(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c24(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c25(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c26(FROM(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c27(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c28(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(first1(z0, z1)) → c29(FIRST1(active(z0), z1), ACTIVE(z0))
ACTIVE(first1(z0, z1)) → c30(FIRST1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons1(z0, z1)) → c31(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c32(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s1(z0)) → c33(S1(active(z0)), ACTIVE(z0))
ACTIVE(unquote(z0)) → c34(UNQUOTE(active(z0)), ACTIVE(z0))
ACTIVE(unquote1(z0)) → c35(UNQUOTE1(active(z0)), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c36(FCONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fcons(z0, z1)) → c37(FCONS(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c4(S(z0))
ACTIVE(quote(s(z0))) → c12(QUOTE(z0))
ACTIVE(unquote(s1(z0))) → c16(UNQUOTE(z0))
ACTIVE(first1(s(z0), cons(x1, mark(z1)))) → c8(CONS1(quote(x1), mark(first1(z0, z1))), QUOTE(x1), FIRST1(z0, mark(z1)))
ACTIVE(first1(s(x0), cons(ok(z0), x2))) → c8(CONS1(ok(quote(z0)), first1(x0, x2)), QUOTE(ok(z0)), FIRST1(x0, x2))
PROPER(sel(z0, z1)) → c68(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c69(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c70(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(first(z0, z1)) → c72(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c74(FROM(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c75(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c76(QUOTE(proper(z0)), PROPER(z0))
PROPER(first1(z0, z1)) → c77(FIRST1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(cons1(z0, z1)) → c79(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote1(z0)) → c81(QUOTE1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c82(S1(proper(z0)), PROPER(z0))
PROPER(unquote(z0)) → c83(UNQUOTE(proper(z0)), PROPER(z0))
PROPER(unquote1(z0)) → c84(UNQUOTE1(proper(z0)), PROPER(z0))
PROPER(fcons(z0, z1)) → c85(FCONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(sel(z0, z1))) → c88(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(s(z0))) → c88(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c88(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(first(z0, z1))) → c88(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(from(z0))) → c88(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(sel1(z0, z1))) → c88(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c88(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(first1(z0, z1))) → c88(TOP(first1(proper(z0), proper(z1))), PROPER(first1(z0, z1)))
TOP(mark(cons1(z0, z1))) → c88(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(quote1(z0))) → c88(TOP(quote1(proper(z0))), PROPER(quote1(z0)))
TOP(mark(s1(z0))) → c88(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(unquote(z0))) → c88(TOP(unquote(proper(z0))), PROPER(unquote(z0)))
TOP(mark(unquote1(z0))) → c88(TOP(unquote1(proper(z0))), PROPER(unquote1(z0)))
TOP(mark(fcons(z0, z1))) → c88(TOP(fcons(proper(z0), proper(z1))), PROPER(fcons(z0, z1)))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(sel(0, cons(z0, z1)))) → c89(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(first(0, z0))) → c89(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c89(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c89(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(sel1(0, cons(z0, z1)))) → c89(TOP(mark(quote(z0))), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(first1(0, z0))) → c89(TOP(mark(nil1)), ACTIVE(first1(0, z0)))
TOP(ok(first1(s(z0), cons(z1, z2)))) → c89(TOP(mark(cons1(quote(z1), first1(z0, z2)))), ACTIVE(first1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c89(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote1(cons(z0, z1)))) → c89(TOP(mark(cons1(quote(z0), quote1(z1)))), ACTIVE(quote1(cons(z0, z1))))
TOP(ok(quote1(nil))) → c89(TOP(mark(nil1)), ACTIVE(quote1(nil)))
TOP(ok(quote(s(z0)))) → c89(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(sel(z0, z1)))) → c89(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(quote1(first(z0, z1)))) → c89(TOP(mark(first1(z0, z1))), ACTIVE(quote1(first(z0, z1))))
TOP(ok(unquote(01))) → c89(TOP(mark(0)), ACTIVE(unquote(01)))
TOP(ok(unquote(s1(z0)))) → c89(TOP(mark(s(unquote(z0)))), ACTIVE(unquote(s1(z0))))
TOP(ok(unquote1(nil1))) → c89(TOP(mark(nil)), ACTIVE(unquote1(nil1)))
TOP(ok(unquote1(cons1(z0, z1)))) → c89(TOP(mark(fcons(unquote(z0), unquote1(z1)))), ACTIVE(unquote1(cons1(z0, z1))))
TOP(ok(fcons(z0, z1))) → c89(TOP(mark(cons(z0, z1))), ACTIVE(fcons(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c89(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(s(z0))) → c89(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c89(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c89(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(from(z0))) → c89(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c89(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(active(z0), z1)), ACTIVE(first1(z0, z1)))
TOP(ok(first1(z0, z1))) → c89(TOP(first1(z0, active(z1))), ACTIVE(first1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c89(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))
TOP(ok(s1(z0))) → c89(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(unquote(z0))) → c89(TOP(unquote(active(z0))), ACTIVE(unquote(z0)))
TOP(ok(unquote1(z0))) → c89(TOP(unquote1(active(z0))), ACTIVE(unquote1(z0)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(active(z0), z1)), ACTIVE(fcons(z0, z1)))
TOP(ok(fcons(z0, z1))) → c89(TOP(fcons(z0, active(z1))), ACTIVE(fcons(z0, z1)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))
TOP(ok(x0)) → c89

S tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))
TOP(ok(x0)) → c89

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1, TOP

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87, c88, c89

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

Removed 5 trailing nodes:

TOP(mark(0)) → c88(TOP(ok(0)))
TOP(mark(nil)) → c88(TOP(ok(nil)))
TOP(ok(x0)) → c89
TOP(mark(nil1)) → c88(TOP(ok(nil1)))
TOP(mark(01)) → c88(TOP(ok(01)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

K tuples:none
Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = x₂
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = [2]x₁
POL(FROM(x₁)) = [4]x₁
POL(QUOTE(x₁)) = x₁
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = [4]x₁
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [2] + x₁

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [4]x₂
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [5]
POL(ok(x₁)) = [2] + x₁

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [2]x₂
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [3]x₁
POL(QUOTE1(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [4] + x₁

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [2]x₂
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [2]x₁
POL(QUOTE1(x₁)) = [2]x₁
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [2]x₁
POL(SEL1(x₁, x₂)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [2] + x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = x₁ + [2]x₂
POL(CONS1(x₁, x₂)) = [4]x₁
POL(FCONS(x₁, x₂)) = x₁ + [4]x₂
POL(FIRST(x₁, x₂)) = x₁
POL(FIRST1(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [3]x₁
POL(QUOTE1(x₁)) = [3]x₁
POL(S(x₁)) = 0
POL(S1(x₁)) = [2]x₁
POL(SEL(x₁, x₂)) = x₁
POL(SEL1(x₁, x₂)) = [4]x₁
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [4] + x₁

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

SEL(z0, mark(z1)) → c39(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

(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.

SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = [4]x₂
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [4]x₁
POL(QUOTE1(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [4]x₁ + [2]x₂
POL(SEL1(x₁, x₂)) = [3]x₁
POL(UNQUOTE(x₁)) = [2]x₁
POL(UNQUOTE1(x₁)) = [4]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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = x₁

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

(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.

SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [4]x₁
POL(CONS1(x₁, x₂)) = x₂
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = [4]x₁
POL(FROM(x₁)) = [3]x₁
POL(QUOTE(x₁)) = [3]x₁
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = [3]x₁
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = x₂
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [2]x₂
POL(CONS1(x₁, x₂)) = [4]x₁ + [4]x₂
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = [2]x₁
POL(FIRST1(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [4]x₁
POL(QUOTE1(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [4]x₁
POL(SEL1(x₁, x₂)) = [4]x₁
POL(UNQUOTE(x₁)) = [2]x₁
POL(UNQUOTE1(x₁)) = [2]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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [4] + x₁

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

S(mark(z0)) → c41(S(z0))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = [3]x₁
POL(FIRST1(x₁, x₂)) = [4]x₂
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [2]x₁
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [4]x₂
POL(SEL1(x₁, x₂)) = 0
POL(UNQUOTE(x₁)) = [4]x₁
POL(UNQUOTE1(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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

S(ok(z0)) → c42(S(z0))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
S(mark(z0)) → c41(S(z0))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

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

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

S(ok(z0)) → c42(S(z0))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [3]x₂
POL(CONS1(x₁, x₂)) = [2]x₁
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = [2]x₁
POL(FIRST1(x₁, x₂)) = [4]x₁
POL(FROM(x₁)) = [4]x₁
POL(QUOTE(x₁)) = [2]x₁
POL(QUOTE1(x₁)) = [4]x₁
POL(S(x₁)) = [4]x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [4]x₂
POL(SEL1(x₁, x₂)) = [2]x₁ + [4]x₂
POL(UNQUOTE(x₁)) = x₁
POL(UNQUOTE1(x₁)) = [4]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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [4] + x₁

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:

FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))

K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

(43) 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.

FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))

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

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

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

POL(CONS(x₁, x₂)) = [4]x₂
POL(CONS1(x₁, x₂)) = [2]x₁ + [2]x₂
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = x₂
POL(FIRST1(x₁, x₂)) = [2]x₁ + [2]x₂
POL(FROM(x₁)) = x₁
POL(QUOTE(x₁)) = [4]x₁
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = [2]x₁
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = [4]x₁ + [2]x₂
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c86(x₁)) = x₁
POL(c87(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(from(z0)) → from(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(first1(z0, z1)) → first1(active(z0), z1)
active(first1(z0, z1)) → first1(z0, active(z1))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
active(s1(z0)) → s1(active(z0))
active(unquote(z0)) → unquote(active(z0))
active(unquote1(z0)) → unquote1(active(z0))
active(fcons(z0, z1)) → fcons(active(z0), z1)
active(fcons(z0, z1)) → fcons(z0, active(z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(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))
first1(mark(z0), z1) → mark(first1(z0, z1))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
unquote(mark(z0)) → mark(unquote(z0))
unquote(ok(z0)) → ok(unquote(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(z0))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
proper(first1(z0, z1)) → first1(proper(z0), proper(z1))
proper(nil1) → ok(nil1)
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
proper(01) → ok(01)
proper(quote1(z0)) → quote1(proper(z0))
proper(s1(z0)) → s1(proper(z0))
proper(unquote(z0)) → unquote(proper(z0))
proper(unquote1(z0)) → unquote1(proper(z0))
proper(fcons(z0, z1)) → fcons(proper(z0), proper(z1))
quote(ok(z0)) → ok(quote(z0))
quote1(ok(z0)) → ok(quote1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))

S tuples:none
K tuples:

FROM(mark(z0)) → c48(FROM(z0))
FROM(ok(z0)) → c49(FROM(z0))
SEL1(mark(z0), z1) → c50(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c52(SEL1(z0, z1))
FIRST1(mark(z0), z1) → c53(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c55(FIRST1(z0, z1))
UNQUOTE1(mark(z0)) → c63(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c64(UNQUOTE1(z0))
FCONS(z0, mark(z1)) → c66(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c67(FCONS(z0, z1))
QUOTE(ok(z0)) → c86(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c44(CONS(z0, z1))
QUOTE1(ok(z0)) → c87(QUOTE1(z0))
SEL(ok(z0), ok(z1)) → c40(SEL(z0, z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
CONS(mark(z0), z1) → c43(CONS(z0, z1))
FIRST(mark(z0), z1) → c45(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c47(FIRST(z0, z1))
FIRST1(z0, mark(z1)) → c54(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c56(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c58(CONS1(z0, z1))
S1(mark(z0)) → c59(S1(z0))
S1(ok(z0)) → c60(S1(z0))
FCONS(mark(z0), z1) → c65(FCONS(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
UNQUOTE(mark(z0)) → c61(UNQUOTE(z0))
SEL1(z0, mark(z1)) → c51(SEL1(z0, z1))
CONS1(z0, mark(z1)) → c57(CONS1(z0, z1))
UNQUOTE(ok(z0)) → c62(UNQUOTE(z0))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
FIRST(z0, mark(z1)) → c46(FIRST(z0, z1))

Defined Rule Symbols:

active, sel, s, cons, first, from, sel1, first1, cons1, s1, unquote, unquote1, fcons, proper, quote, quote1, top

Defined Pair Symbols:

SEL, S, CONS, FIRST, FROM, SEL1, FIRST1, CONS1, S1, UNQUOTE, UNQUOTE1, FCONS, QUOTE, QUOTE1

Compound Symbols:

c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c86, c87

(45) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtUnreachableProof (EQUIVALENT transformation)

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

(19) CdtUnreachableProof (EQUIVALENT transformation)

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

(45) SIsEmptyProof (EQUIVALENT transformation)

(46) BOUNDS(O(1), O(1))