Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    app(app(times,x),app(app(plus,y),app(s,z)))  → app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z)))
2:    app(app(times,x),0)  → 0
3:    app(app(times,x),app(s,y))  → app(app(plus,app(app(times,x),y)),x)
4:    app(app(plus,x),0)  → x
5:    app(app(plus,x),app(s,y))  → app(s,app(app(plus,x),y))
There are 12 dependency pairs:
6:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z)))
7:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0))))
8:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))
9:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(app(plus,y),app(app(times,app(s,z)),0))
10:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(app(times,app(s,z)),0)
11:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(times,app(s,z))
12:    APP(app(times,x),app(app(plus,y),app(s,z)))  → APP(app(times,x),app(s,z))
13:    APP(app(times,x),app(s,y))  → APP(app(plus,app(app(times,x),y)),x)
14:    APP(app(times,x),app(s,y))  → APP(plus,app(app(times,x),y))
15:    APP(app(times,x),app(s,y))  → APP(app(times,x),y)
16:    APP(app(plus,x),app(s,y))  → APP(s,app(app(plus,x),y))
17:    APP(app(plus,x),app(s,y))  → APP(app(plus,x),y)
The approximated dependency graph contains one SCC: {6,8,9,12,13,15,17}.
Tyrolean Termination Tool  (0.18 seconds)   ---  May 3, 2006