Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    le(0,y)  → true
2:    le(s(x),0)  → false
3:    le(s(x),s(y))  → le(x,y)
4:    minus(x,0)  → x
5:    minus(s(x),s(y))  → minus(x,y)
6:    gcd(0,y)  → y
7:    gcd(s(x),0)  → s(x)
8:    gcd(s(x),s(y))  → if_gcd(le(y,x),s(x),s(y))
9:    if_gcd(true,s(x),s(y))  → gcd(minus(x,y),s(y))
10:    if_gcd(false,s(x),s(y))  → gcd(minus(y,x),s(x))
There are 8 dependency pairs:
11:    LE(s(x),s(y))  → LE(x,y)
12:    MINUS(s(x),s(y))  → MINUS(x,y)
13:    GCD(s(x),s(y))  → IF_GCD(le(y,x),s(x),s(y))
14:    GCD(s(x),s(y))  → LE(y,x)
15:    IF_GCD(true,s(x),s(y))  → GCD(minus(x,y),s(y))
16:    IF_GCD(true,s(x),s(y))  → MINUS(x,y)
17:    IF_GCD(false,s(x),s(y))  → GCD(minus(y,x),s(x))
18:    IF_GCD(false,s(x),s(y))  → MINUS(y,x)
The approximated dependency graph contains 3 SCCs: {11}, {12} and {13,15,17}.
Tyrolean Termination Tool  (0.25 seconds)   ---  May 3, 2006