Term Rewriting System R:

   [X, Y, Z]
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   sel(0, cons(X, Z)) -> X
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))
   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
   sel1(0, cons(X, Z)) -> quote(X)
   first1(0, Z) -> nil1
   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
   quote(0) -> 01
   quote(s(X)) -> s1(quote(X))
   quote(sel(X, Z)) -> sel1(X, Z)
   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
   quote1(nil) -> nil1
   quote1(first(X, Z)) -> first1(X, Z)
   unquote(01) -> 0
   unquote(s1(X)) -> s(unquote(X))
   unquote1(nil1) -> nil
   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
   fcons(X, Z) -> cons(X, Z)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)
   UNQUOTE1(cons1(X, Z)) -> FCONS(unquote(X), unquote1(Z))
   UNQUOTE1(cons1(X, Z)) -> UNQUOTE(X)
   UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)
   FROM(X) -> FROM(s(X))
   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
   QUOTE1(cons(X, Z)) -> QUOTE(X)
   QUOTE1(cons(X, Z)) -> QUOTE1(Z)
   QUOTE1(first(X, Z)) -> FIRST1(X, Z)
   QUOTE(sel(X, Z)) -> SEL1(X, Z)
   QUOTE(s(X)) -> QUOTE(X)
   FIRST1(s(X), cons(Y, Z)) -> QUOTE(Y)
   FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)
   UNQUOTE(s1(X)) -> UNQUOTE(X)
   SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)
   SEL1(0, cons(X, Z)) -> QUOTE(X)

Furthermore, R contains eight SCCs.


SCC1:

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

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)) = 1 + x_1
POL(FIRST(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)



 This transformation is resulting in no new subcycles.


SCC2:

UNQUOTE(s1(X)) -> UNQUOTE(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(s1(x_1)) = 1 + x_1
POL(UNQUOTE(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   UNQUOTE(s1(X)) -> UNQUOTE(X)



 This transformation is resulting in no new subcycles.


SCC3:

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: 

   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   unquote1(nil1) -> nil
   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
   fcons(X, Z) -> cons(X, Z)
   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
   quote1(nil) -> nil1
   quote1(first(X, Z)) -> first1(X, Z)
   quote(0) -> 01
   quote(sel(X, Z)) -> sel1(X, Z)
   quote(s(X)) -> s1(quote(X))
   first1(0, Z) -> nil1
   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
   unquote(s1(X)) -> s(unquote(X))
   unquote(01) -> 0
   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
   sel1(0, cons(X, Z)) -> quote(X)


 This transformation is resulting in one new subcycle:


SCC3.MRR1:

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

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC3.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: 1.14 seconds.