Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    app(app(plus,0),y)  → y
2:    app(app(plus,app(s,x)),y)  → app(s,app(app(plus,x),y))
3:    app(app(times,0),y)  → 0
4:    app(app(times,app(s,x)),y)  → app(app(plus,app(app(times,x),y)),y)
5:    app(app(map,f),nil)  → nil
6:    app(app(map,f),app(app(cons,x),xs))  → app(app(cons,app(f,x)),app(app(map,f),xs))
7:    inc  → app(map,app(plus,app(s,0)))
8:    double  → app(map,app(times,app(s,app(s,0))))
There are 18 dependency pairs:
9:    APP(app(plus,app(s,x)),y)  → APP(s,app(app(plus,x),y))
10:    APP(app(plus,app(s,x)),y)  → APP(app(plus,x),y)
11:    APP(app(plus,app(s,x)),y)  → APP(plus,x)
12:    APP(app(times,app(s,x)),y)  → APP(app(plus,app(app(times,x),y)),y)
13:    APP(app(times,app(s,x)),y)  → APP(plus,app(app(times,x),y))
14:    APP(app(times,app(s,x)),y)  → APP(app(times,x),y)
15:    APP(app(times,app(s,x)),y)  → APP(times,x)
16:    APP(app(map,f),app(app(cons,x),xs))  → APP(app(cons,app(f,x)),app(app(map,f),xs))
17:    APP(app(map,f),app(app(cons,x),xs))  → APP(cons,app(f,x))
18:    APP(app(map,f),app(app(cons,x),xs))  → APP(f,x)
19:    APP(app(map,f),app(app(cons,x),xs))  → APP(app(map,f),xs)
20:    INC  → APP(map,app(plus,app(s,0)))
21:    INC  → APP(plus,app(s,0))
22:    INC  → APP(s,0)
23:    DOUBLE  → APP(map,app(times,app(s,app(s,0))))
24:    DOUBLE  → APP(times,app(s,app(s,0)))
25:    DOUBLE  → APP(s,app(s,0))
26:    DOUBLE  → APP(s,0)
The approximated dependency graph contains one SCC: {10,12,14,16,18,19}.
Tyrolean Termination Tool  (0.15 seconds)   ---  May 3, 2006