Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    app(app(app(if,true),x),y)  → x
2:    app(app(app(if,true),x),y)  → y
3:    app(app(takeWhile,p),nil)  → nil
4:    app(app(takeWhile,p),app(app(cons,x),xs))  → app(app(app(if,app(p,x)),app(app(cons,x),app(app(takeWhile,p),xs))),nil)
5:    app(app(dropWhile,p),nil)  → nil
6:    app(app(dropWhile,p),app(app(cons,x),xs))  → app(app(app(if,app(p,x)),app(app(dropWhile,p),xs)),app(app(cons,x),xs))
There are 11 dependency pairs:
7:    APP(app(takeWhile,p),app(app(cons,x),xs))  → APP(app(app(if,app(p,x)),app(app(cons,x),app(app(takeWhile,p),xs))),nil)
8:    APP(app(takeWhile,p),app(app(cons,x),xs))  → APP(app(if,app(p,x)),app(app(cons,x),app(app(takeWhile,p),xs)))
9:    APP(app(takeWhile,p),app(app(cons,x),xs))  → APP(if,app(p,x))
10:    APP(app(takeWhile,p),app(app(cons,x),xs))  → APP(p,x)
11:    APP(app(takeWhile,p),app(app(cons,x),xs))  → APP(app(cons,x),app(app(takeWhile,p),xs))
12:    APP(app(takeWhile,p),app(app(cons,x),xs))  → APP(app(takeWhile,p),xs)
13:    APP(app(dropWhile,p),app(app(cons,x),xs))  → APP(app(app(if,app(p,x)),app(app(dropWhile,p),xs)),app(app(cons,x),xs))
14:    APP(app(dropWhile,p),app(app(cons,x),xs))  → APP(app(if,app(p,x)),app(app(dropWhile,p),xs))
15:    APP(app(dropWhile,p),app(app(cons,x),xs))  → APP(if,app(p,x))
16:    APP(app(dropWhile,p),app(app(cons,x),xs))  → APP(p,x)
17:    APP(app(dropWhile,p),app(app(cons,x),xs))  → APP(app(dropWhile,p),xs)
The approximated dependency graph contains one SCC: {8,10-14,16,17}.
Tyrolean Termination Tool  (0.11 seconds)   ---  May 3, 2006