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(app(quot,0),app(s,y))  → 0
4:    app(app(quot,app(s,x)),app(s,y))  → app(s,app(app(quot,app(app(minus,x),y)),app(s,y)))
5:    app(log,app(s,0))  → 0
6:    app(log,app(s,app(s,x)))  → app(s,app(log,app(s,app(app(quot,x),app(s,app(s,0))))))
There are 14 dependency pairs:
7:    APP(app(minus,app(s,x)),app(s,y))  → APP(app(minus,x),y)
8:    APP(app(minus,app(s,x)),app(s,y))  → APP(minus,x)
9:    APP(app(quot,app(s,x)),app(s,y))  → APP(s,app(app(quot,app(app(minus,x),y)),app(s,y)))
10:    APP(app(quot,app(s,x)),app(s,y))  → APP(app(quot,app(app(minus,x),y)),app(s,y))
11:    APP(app(quot,app(s,x)),app(s,y))  → APP(quot,app(app(minus,x),y))
12:    APP(app(quot,app(s,x)),app(s,y))  → APP(app(minus,x),y)
13:    APP(app(quot,app(s,x)),app(s,y))  → APP(minus,x)
14:    APP(log,app(s,app(s,x)))  → APP(s,app(log,app(s,app(app(quot,x),app(s,app(s,0))))))
15:    APP(log,app(s,app(s,x)))  → APP(log,app(s,app(app(quot,x),app(s,app(s,0)))))
16:    APP(log,app(s,app(s,x)))  → APP(s,app(app(quot,x),app(s,app(s,0))))
17:    APP(log,app(s,app(s,x)))  → APP(app(quot,x),app(s,app(s,0)))
18:    APP(log,app(s,app(s,x)))  → APP(quot,x)
19:    APP(log,app(s,app(s,x)))  → APP(s,app(s,0))
20:    APP(log,app(s,app(s,x)))  → APP(s,0)
The approximated dependency graph contains one SCC: {7,10,12,15,17}.
Tyrolean Termination Tool  (0.16 seconds)   ---  May 3, 2006