Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    app(app(map,f),nil)  → nil
2:    app(app(map,f),app(app(cons,x),xs))  → app(app(cons,app(f,x)),app(app(map,f),xs))
3:    app(sum,app(app(cons,x),xs))  → app(app(plus,x),app(sum,xs))
4:    app(size,app(app(node,x),xs))  → app(s,app(sum,app(app(map,size),xs)))
5:    app(app(plus,0),x)  → 0
6:    app(app(plus,app(s,x)),y)  → app(s,app(app(plus,x),y))
There are 14 dependency pairs:
7:    APP(app(map,f),app(app(cons,x),xs))  → APP(app(cons,app(f,x)),app(app(map,f),xs))
8:    APP(app(map,f),app(app(cons,x),xs))  → APP(cons,app(f,x))
9:    APP(app(map,f),app(app(cons,x),xs))  → APP(f,x)
10:    APP(app(map,f),app(app(cons,x),xs))  → APP(app(map,f),xs)
11:    APP(sum,app(app(cons,x),xs))  → APP(app(plus,x),app(sum,xs))
12:    APP(sum,app(app(cons,x),xs))  → APP(plus,x)
13:    APP(sum,app(app(cons,x),xs))  → APP(sum,xs)
14:    APP(size,app(app(node,x),xs))  → APP(s,app(sum,app(app(map,size),xs)))
15:    APP(size,app(app(node,x),xs))  → APP(sum,app(app(map,size),xs))
16:    APP(size,app(app(node,x),xs))  → APP(app(map,size),xs)
17:    APP(size,app(app(node,x),xs))  → APP(map,size)
18:    APP(app(plus,app(s,x)),y)  → APP(s,app(app(plus,x),y))
19:    APP(app(plus,app(s,x)),y)  → APP(app(plus,x),y)
20:    APP(app(plus,app(s,x)),y)  → APP(plus,x)
The approximated dependency graph contains one SCC: {7,9-11,13,15,16,19}.
Tyrolean Termination Tool  (2.41 seconds)   ---  May 3, 2006