Term Rewriting System R:
[X, Y, Z, X1, X2]
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(dbl(s(X))) -> S(s(dbl(X)))
ACTIVE(dbl(s(X))) -> S(dbl(X))
ACTIVE(dbl(s(X))) -> DBL(X)
ACTIVE(dbls(cons(X, Y))) -> CONS(dbl(X), dbls(Y))
ACTIVE(dbls(cons(X, Y))) -> DBL(X)
ACTIVE(dbls(cons(X, Y))) -> DBLS(Y)
ACTIVE(sel(s(X), cons(Y, Z))) -> SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) -> CONS(sel(X, Z), indx(Y, Z))
ACTIVE(indx(cons(X, Y), Z)) -> SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) -> INDX(Y, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(dbl1(s(X))) -> S1(s1(dbl1(X)))
ACTIVE(dbl1(s(X))) -> S1(dbl1(X))
ACTIVE(dbl1(s(X))) -> DBL1(X)
ACTIVE(sel1(s(X), cons(Y, Z))) -> SEL1(X, Z)
ACTIVE(quote(s(X))) -> S1(quote(X))
ACTIVE(quote(s(X))) -> QUOTE(X)
ACTIVE(quote(dbl(X))) -> DBL1(X)
ACTIVE(quote(sel(X, Y))) -> SEL1(X, Y)
ACTIVE(dbl(X)) -> DBL(active(X))
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(dbls(X)) -> DBLS(active(X))
ACTIVE(dbls(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(indx(X1, X2)) -> INDX(active(X1), X2)
ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl1(X)) -> DBL1(active(X))
ACTIVE(dbl1(X)) -> ACTIVE(X)
ACTIVE(s1(X)) -> S1(active(X))
ACTIVE(s1(X)) -> ACTIVE(X)
ACTIVE(sel1(X1, X2)) -> SEL1(active(X1), X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel1(X1, X2)) -> SEL1(X1, active(X2))
ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(quote(X)) -> QUOTE(active(X))
ACTIVE(quote(X)) -> ACTIVE(X)
DBL(mark(X)) -> DBL(X)
DBL(ok(X)) -> DBL(X)
DBLS(mark(X)) -> DBLS(X)
DBLS(ok(X)) -> DBLS(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
INDX(mark(X1), X2) -> INDX(X1, X2)
INDX(ok(X1), ok(X2)) -> INDX(X1, X2)
DBL1(mark(X)) -> DBL1(X)
DBL1(ok(X)) -> DBL1(X)
S1(mark(X)) -> S1(X)
S1(ok(X)) -> S1(X)
SEL1(mark(X1), X2) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)
SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
QUOTE(mark(X)) -> QUOTE(X)
QUOTE(ok(X)) -> QUOTE(X)
PROPER(dbl(X)) -> DBL(proper(X))
PROPER(dbl(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(dbls(X)) -> DBLS(proper(X))
PROPER(dbls(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(indx(X1, X2)) -> INDX(proper(X1), proper(X2))
PROPER(indx(X1, X2)) -> PROPER(X1)
PROPER(indx(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(dbl1(X)) -> DBL1(proper(X))
PROPER(dbl1(X)) -> PROPER(X)
PROPER(s1(X)) -> S1(proper(X))
PROPER(s1(X)) -> PROPER(X)
PROPER(sel1(X1, X2)) -> SEL1(proper(X1), proper(X2))
PROPER(sel1(X1, X2)) -> PROPER(X1)
PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(quote(X)) -> QUOTE(proper(X))
PROPER(quote(X)) -> PROPER(X)
S(ok(X)) -> S(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(ok(X)) -> FROM(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains 14 SCCs.


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)
       →DP Problem 5
Remaining Obligation(s)
       →DP Problem 6
Remaining Obligation(s)
       →DP Problem 7
Remaining Obligation(s)
       →DP Problem 8
Remaining Obligation(s)
       →DP Problem 9
Remaining Obligation(s)
       →DP Problem 10
Remaining Obligation(s)
       →DP Problem 11
Remaining Obligation(s)
       →DP Problem 12
Remaining Obligation(s)
       →DP Problem 13
Remaining Obligation(s)
       →DP Problem 14
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)
       →DP Problem 5
Remaining Obligation(s)
       →DP Problem 6
Remaining Obligation(s)
       →DP Problem 7
Remaining Obligation(s)
       →DP Problem 8
Remaining Obligation(s)
       →DP Problem 9
Remaining Obligation(s)
       →DP Problem 10
Remaining Obligation(s)
       →DP Problem 11
Remaining Obligation(s)
       →DP Problem 12
Remaining Obligation(s)
       →DP Problem 13
Remaining Obligation(s)
       →DP Problem 14
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)
       →DP Problem 5
Remaining Obligation(s)
       →DP Problem 6
Remaining Obligation(s)
       →DP Problem 7
Remaining Obligation(s)
       →DP Problem 8
Remaining Obligation(s)
       →DP Problem 9
Remaining Obligation(s)
       →DP Problem 10
Remaining Obligation(s)
       →DP Problem 11
Remaining Obligation(s)
       →DP Problem 12
Remaining Obligation(s)
       →DP Problem 13
Remaining Obligation(s)
       →DP Problem 14
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)
       →DP Problem 5
Remaining Obligation(s)
       →DP Problem 6
Remaining Obligation(s)
       →DP Problem 7
Remaining Obligation(s)
       →DP Problem 8
Remaining Obligation(s)
       →DP Problem 9
Remaining Obligation(s)
       →DP Problem 10
Remaining Obligation(s)
       →DP Problem 11
Remaining Obligation(s)
       →DP Problem 12
Remaining Obligation(s)
       →DP Problem 13
Remaining Obligation(s)
       →DP Problem 14
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)
       →DP Problem 5
Remaining Obligation(s)
       →DP Problem 6
Remaining Obligation(s)
       →DP Problem 7
Remaining Obligation(s)
       →DP Problem 8
Remaining Obligation(s)
       →DP Problem 9
Remaining Obligation(s)
       →DP Problem 10
Remaining Obligation(s)
       →DP Problem 11
Remaining Obligation(s)
       →DP Problem 12
Remaining Obligation(s)
       →DP Problem 13
Remaining Obligation(s)
       →DP Problem 14
Remaining Obligation(s)




The following remains to be proven: