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'(minus,app'(app'(minus,x),y)),z)  → app'(app'(minus,x),app'(app'(plus,y),z))
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)))
6:    app'(app'(plus,0),y)  → y
7:    app'(app'(plus,app'(s,x)),y)  → app'(s,app'(app'(plus,x),y))
8:    app'(app'(app,nil),k)  → k
9:    app'(app'(app,l),nil)  → l
10:    app'(app'(app,app'(app'(cons,x),l)),k)  → app'(app'(cons,x),app'(app'(app,l),k))
11:    app'(sum,app'(app'(cons,x),nil))  → app'(app'(cons,x),nil)
12:    app'(sum,app'(app'(cons,x),app'(app'(cons,y),l)))  → app'(sum,app'(app'(cons,app'(app'(plus,x),y)),l))
13:    app'(sum,app'(app'(app,l),app'(app'(cons,x),app'(app'(cons,y),k))))  → app'(sum,app'(app'(app,l),app'(sum,app'(app'(cons,x),app'(app'(cons,y),k)))))
There are 24 dependency pairs:
14:    APP'(app'(minus,app'(s,x)),app'(s,y))  → APP'(app'(minus,x),y)
15:    APP'(app'(minus,app'(s,x)),app'(s,y))  → APP'(minus,x)
16:    APP'(app'(minus,app'(app'(minus,x),y)),z)  → APP'(app'(minus,x),app'(app'(plus,y),z))
17:    APP'(app'(minus,app'(app'(minus,x),y)),z)  → APP'(app'(plus,y),z)
18:    APP'(app'(minus,app'(app'(minus,x),y)),z)  → APP'(plus,y)
19:    APP'(app'(quot,app'(s,x)),app'(s,y))  → APP'(s,app'(app'(quot,app'(app'(minus,x),y)),app'(s,y)))
20:    APP'(app'(quot,app'(s,x)),app'(s,y))  → APP'(app'(quot,app'(app'(minus,x),y)),app'(s,y))
21:    APP'(app'(quot,app'(s,x)),app'(s,y))  → APP'(quot,app'(app'(minus,x),y))
22:    APP'(app'(quot,app'(s,x)),app'(s,y))  → APP'(app'(minus,x),y)
23:    APP'(app'(quot,app'(s,x)),app'(s,y))  → APP'(minus,x)
24:    APP'(app'(plus,app'(s,x)),y)  → APP'(s,app'(app'(plus,x),y))
25:    APP'(app'(plus,app'(s,x)),y)  → APP'(app'(plus,x),y)
26:    APP'(app'(plus,app'(s,x)),y)  → APP'(plus,x)
27:    APP'(app'(app,app'(app'(cons,x),l)),k)  → APP'(app'(cons,x),app'(app'(app,l),k))
28:    APP'(app'(app,app'(app'(cons,x),l)),k)  → APP'(app'(app,l),k)
29:    APP'(app'(app,app'(app'(cons,x),l)),k)  → APP'(app,l)
30:    APP'(sum,app'(app'(cons,x),app'(app'(cons,y),l)))  → APP'(sum,app'(app'(cons,app'(app'(plus,x),y)),l))
31:    APP'(sum,app'(app'(cons,x),app'(app'(cons,y),l)))  → APP'(app'(cons,app'(app'(plus,x),y)),l)
32:    APP'(sum,app'(app'(cons,x),app'(app'(cons,y),l)))  → APP'(cons,app'(app'(plus,x),y))
33:    APP'(sum,app'(app'(cons,x),app'(app'(cons,y),l)))  → APP'(app'(plus,x),y)
34:    APP'(sum,app'(app'(cons,x),app'(app'(cons,y),l)))  → APP'(plus,x)
35:    APP'(sum,app'(app'(app,l),app'(app'(cons,x),app'(app'(cons,y),k))))  → APP'(sum,app'(app'(app,l),app'(sum,app'(app'(cons,x),app'(app'(cons,y),k)))))
36:    APP'(sum,app'(app'(app,l),app'(app'(cons,x),app'(app'(cons,y),k))))  → APP'(app'(app,l),app'(sum,app'(app'(cons,x),app'(app'(cons,y),k))))
37:    APP'(sum,app'(app'(app,l),app'(app'(cons,x),app'(app'(cons,y),k))))  → APP'(sum,app'(app'(cons,x),app'(app'(cons,y),k)))
The approximated dependency graph contains one SCC: {14,16,17,20,22,25,27,28,30,31,33,35-37}.
Tyrolean Termination Tool  (11.64 seconds)   ---  May 3, 2006