Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    app(pred,app(s,x))  → x
2:    app(app(minus,x),0)  → x
3:    app(app(minus,x),app(s,y))  → app(pred,app(app(minus,x),y))
4:    app(app(quot,0),app(s,y))  → 0
5:    app(app(quot,app(s,x)),app(s,y))  → app(s,app(app(quot,app(app(minus,x),y)),app(s,y)))
There are 7 dependency pairs:
6:    APP(app(minus,x),app(s,y))  → APP(pred,app(app(minus,x),y))
7:    APP(app(minus,x),app(s,y))  → APP(app(minus,x),y)
8:    APP(app(quot,app(s,x)),app(s,y))  → APP(s,app(app(quot,app(app(minus,x),y)),app(s,y)))
9:    APP(app(quot,app(s,x)),app(s,y))  → APP(app(quot,app(app(minus,x),y)),app(s,y))
10:    APP(app(quot,app(s,x)),app(s,y))  → APP(quot,app(app(minus,x),y))
11:    APP(app(quot,app(s,x)),app(s,y))  → APP(app(minus,x),y)
12:    APP(app(quot,app(s,x)),app(s,y))  → APP(minus,x)
The approximated dependency graph contains 2 SCCs: {7} and {9,11}.
Tyrolean Termination Tool  (0.05 seconds)   ---  May 3, 2006