Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    app(app(minus,x),0)  → x
2:    app(app(minus,app(s,x)),app(s,y))  → app(app(minus,x),y)
3:    app(double,0)  → 0
4:    app(double,app(s,x))  → app(s,app(s,app(double,x)))
5:    app(app(plus,0),y)  → y
6:    app(app(plus,app(s,x)),y)  → app(s,app(app(plus,x),y))
7:    app(app(plus,app(s,x)),y)  → app(app(plus,x),app(s,y))
8:    app(app(plus,app(s,x)),y)  → app(s,app(app(plus,app(app(minus,x),y)),app(double,y)))
There are 16 dependency pairs:
9:    APP(app(minus,app(s,x)),app(s,y))  → APP(app(minus,x),y)
10:    APP(app(minus,app(s,x)),app(s,y))  → APP(minus,x)
11:    APP(double,app(s,x))  → APP(s,app(s,app(double,x)))
12:    APP(double,app(s,x))  → APP(s,app(double,x))
13:    APP(double,app(s,x))  → APP(double,x)
14:    APP(app(plus,app(s,x)),y)  → APP(s,app(app(plus,x),y))
15:    APP(app(plus,app(s,x)),y)  → APP(app(plus,x),y)
16:    APP(app(plus,app(s,x)),y)  → APP(app(plus,x),app(s,y))
17:    APP(app(plus,app(s,x)),y)  → APP(plus,x)
18:    APP(app(plus,app(s,x)),y)  → APP(s,y)
19:    APP(app(plus,app(s,x)),y)  → APP(s,app(app(plus,app(app(minus,x),y)),app(double,y)))
20:    APP(app(plus,app(s,x)),y)  → APP(app(plus,app(app(minus,x),y)),app(double,y))
21:    APP(app(plus,app(s,x)),y)  → APP(plus,app(app(minus,x),y))
22:    APP(app(plus,app(s,x)),y)  → APP(app(minus,x),y)
23:    APP(app(plus,app(s,x)),y)  → APP(minus,x)
24:    APP(app(plus,app(s,x)),y)  → APP(double,y)
The approximated dependency graph contains 2 SCCs: {13} and {9,15,16,20,22}.
Tyrolean Termination Tool  (0.05 seconds)   ---  May 3, 2006