Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    dbl(0)  → 0
2:    dbl(s(X))  → s(s(dbl(X)))
3:    dbls(nil)  → nil
4:    dbls(cons(X,Y))  → cons(dbl(X),dbls(Y))
5:    sel(0,cons(X,Y))  → X
6:    sel(s(X),cons(Y,Z))  → sel(X,Z)
7:    indx(nil,X)  → nil
8:    indx(cons(X,Y),Z)  → cons(sel(X,Z),indx(Y,Z))
9:    from(X)  → cons(X,from(s(X)))
10:    dbl1(0)  → 01
11:    dbl1(s(X))  → s1(s1(dbl1(X)))
12:    sel1(0,cons(X,Y))  → X
13:    sel1(s(X),cons(Y,Z))  → sel1(X,Z)
14:    quote(0)  → 01
15:    quote(s(X))  → s1(quote(X))
16:    quote(dbl(X))  → dbl1(X)
17:    quote(sel(X,Y))  → sel1(X,Y)
There are 12 dependency pairs:
18:    DBL(s(X))  → DBL(X)
19:    DBLS(cons(X,Y))  → DBL(X)
20:    DBLS(cons(X,Y))  → DBLS(Y)
21:    SEL(s(X),cons(Y,Z))  → SEL(X,Z)
22:    INDX(cons(X,Y),Z)  → SEL(X,Z)
23:    INDX(cons(X,Y),Z)  → INDX(Y,Z)
24:    FROM(X)  → FROM(s(X))
25:    DBL1(s(X))  → DBL1(X)
26:    SEL1(s(X),cons(Y,Z))  → SEL1(X,Z)
27:    QUOTE(s(X))  → QUOTE(X)
28:    QUOTE(dbl(X))  → DBL1(X)
29:    QUOTE(sel(X,Y))  → SEL1(X,Y)
The approximated dependency graph contains 8 SCCs: {18}, {25}, {20}, {24}, {21}, {23}, {26} and {27}.
Tyrolean Termination Tool  (0.02 seconds)   ---  May 3, 2006