Term Rewriting System R:

   [X, Y, Z]
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   dbls(nil) -> nil
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   indx(nil, X) -> nil
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   from(X) -> cons(X, from(s(X)))
   dbl1(0) -> 01
   dbl1(s(X)) -> s1(s1(dbl1(X)))
   sel1(0, cons(X, Y)) -> X
   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
   quote(0) -> 01
   quote(s(X)) -> s1(quote(X))
   quote(dbl(X)) -> dbl1(X)
   quote(sel(X, Y)) -> sel1(X, Y)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   FROM(X) -> FROM(s(X))
   QUOTE(sel(X, Y)) -> SEL1(X, Y)
   QUOTE(s(X)) -> QUOTE(X)
   QUOTE(dbl(X)) -> DBL1(X)
   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
   SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)
   INDX(cons(X, Y), Z) -> SEL(X, Z)
   INDX(cons(X, Y), Z) -> INDX(Y, Z)
   DBLS(cons(X, Y)) -> DBL(X)
   DBLS(cons(X, Y)) -> DBLS(Y)
   DBL(s(X)) -> DBL(X)
   DBL1(s(X)) -> DBL1(X)

Furthermore, R contains eight SCCs.


SCC1:

FROM(X) -> FROM(s(X))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = x_1
POL(FROM(x_1)) = 1 + x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   from(X) -> cons(X, from(s(X)))
   quote(sel(X, Y)) -> sel1(X, Y)
   quote(0) -> 01
   quote(s(X)) -> s1(quote(X))
   quote(dbl(X)) -> dbl1(X)
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   sel1(0, cons(X, Y)) -> X
   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
   indx(nil, X) -> nil
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   dbls(nil) -> nil
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   dbl1(s(X)) -> s1(s1(dbl1(X)))
   dbl1(0) -> 01


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

FROM(X) -> FROM(s(X))

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.NOC1:

FROM(X) -> FROM(s(X))

Found an infinite P-chain over R:
P = FROM(X) -> FROM(s(X))
R = []
s = FROM(X) evaluates to t = FROM(s(X))
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 0.832 seconds.