Termination Proof Script

Consider the TRS R consisting of the rewrite rules
1:    g(A)  → A
2:    g(B)  → A
3:    g(B)  → B
4:    g(C)  → A
5:    g(C)  → B
6:    g(C)  → C
7:    foldB(t,0)  → t
8:    foldB(t,s(n))  → f(foldB(t,n),B)
9:    foldC(t,0)  → t
10:    foldC(t,s(n))  → f(foldC(t,n),C)
11:    f(t,x)  → f'(t,g(x))
12:    f'(triple(a,b,c),C)  → triple(a,b,s(c))
13:    f'(triple(a,b,c),B)  → f(triple(a,b,c),A)
14:    f'(triple(a,b,c),A)  → f''(foldB(triple(s(a),0,c),b))
15:    f''(triple(a,b,c))  → foldC(triple(a,b,0),c)
16:    fold(t,x,0)  → t
17:    fold(t,x,s(n))  → f(fold(t,x,n),x)
There are 12 dependency pairs:
18:    FOLDB(t,s(n))  → F(foldB(t,n),B)
19:    FOLDB(t,s(n))  → FOLDB(t,n)
20:    FOLDC(t,s(n))  → F(foldC(t,n),C)
21:    FOLDC(t,s(n))  → FOLDC(t,n)
22:    F(t,x)  → F'(t,g(x))
23:    F(t,x)  → G(x)
24:    F'(triple(a,b,c),B)  → F(triple(a,b,c),A)
25:    F'(triple(a,b,c),A)  → F''(foldB(triple(s(a),0,c),b))
26:    F'(triple(a,b,c),A)  → FOLDB(triple(s(a),0,c),b)
27:    F''(triple(a,b,c))  → FOLDC(triple(a,b,0),c)
28:    FOLD(t,x,s(n))  → F(fold(t,x,n),x)
29:    FOLD(t,x,s(n))  → FOLD(t,x,n)
The approximated dependency graph contains 2 SCCs: {18-22,24-27} and {29}.
Tyrolean Termination Tool  (7.36 seconds)   ---  May 3, 2006