Certification Problem
Input (TPDB TRS_Standard/CiME_04/filliatre2)
The rewrite relation of the following TRS is considered.
|
g(A) |
→ |
A |
(1) |
|
g(B) |
→ |
A |
(2) |
|
g(B) |
→ |
B |
(3) |
|
g(C) |
→ |
A |
(4) |
|
g(C) |
→ |
B |
(5) |
|
g(C) |
→ |
C |
(6) |
|
foldB(t,0) |
→ |
t |
(7) |
|
foldB(t,s(n)) |
→ |
f(foldB(t,n),B) |
(8) |
|
foldC(t,0) |
→ |
t |
(9) |
|
foldC(t,s(n)) |
→ |
f(foldC(t,n),C) |
(10) |
|
f(t,x) |
→ |
f'(t,g(x)) |
(11) |
|
f'(triple(a,b,c),C) |
→ |
triple(a,b,s(c)) |
(12) |
|
f'(triple(a,b,c),B) |
→ |
f(triple(a,b,c),A) |
(13) |
|
f'(triple(a,b,c),A) |
→ |
f''(foldB(triple(s(a),0,c),b)) |
(14) |
|
f''(triple(a,b,c)) |
→ |
foldC(triple(a,b,0),c) |
(15) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [g(x1)] |
= |
1 · x1
|
| [A] |
= |
2 |
| [B] |
= |
2 |
| [C] |
= |
2 |
| [foldB(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [0] |
= |
0 |
| [s(x1)] |
= |
2 + 1 · x1
|
| [f(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [foldC(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [f'(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [triple(x1, x2, x3)] |
= |
1 · x1 + 2 · x2 + 2 · x3
|
| [f''(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
f'(triple(a,b,c),A) |
→ |
f''(foldB(triple(s(a),0,c),b)) |
(14) |
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(A) |
= |
10 |
|
weight(A) |
= |
1 |
|
|
|
| prec(B) |
= |
2 |
|
weight(B) |
= |
2 |
|
|
|
| prec(C) |
= |
0 |
|
weight(C) |
= |
3 |
|
|
|
| prec(0) |
= |
3 |
|
weight(0) |
= |
2 |
|
|
|
| prec(g) |
= |
11 |
|
weight(g) |
= |
0 |
|
|
|
| prec(s) |
= |
4 |
|
weight(s) |
= |
3 |
|
|
|
| prec(f'') |
= |
9 |
|
weight(f'') |
= |
3 |
|
|
|
| prec(foldB) |
= |
1 |
|
weight(foldB) |
= |
0 |
|
|
|
| prec(f) |
= |
7 |
|
weight(f) |
= |
0 |
|
|
|
| prec(foldC) |
= |
8 |
|
weight(foldC) |
= |
0 |
|
|
|
| prec(f') |
= |
6 |
|
weight(f') |
= |
0 |
|
|
|
| prec(triple) |
= |
5 |
|
weight(triple) |
= |
0 |
|
|
|
all of the following rules can be deleted.
|
g(A) |
→ |
A |
(1) |
|
g(B) |
→ |
A |
(2) |
|
g(B) |
→ |
B |
(3) |
|
g(C) |
→ |
A |
(4) |
|
g(C) |
→ |
B |
(5) |
|
g(C) |
→ |
C |
(6) |
|
foldB(t,0) |
→ |
t |
(7) |
|
foldB(t,s(n)) |
→ |
f(foldB(t,n),B) |
(8) |
|
foldC(t,0) |
→ |
t |
(9) |
|
foldC(t,s(n)) |
→ |
f(foldC(t,n),C) |
(10) |
|
f(t,x) |
→ |
f'(t,g(x)) |
(11) |
|
f'(triple(a,b,c),C) |
→ |
triple(a,b,s(c)) |
(12) |
|
f'(triple(a,b,c),B) |
→ |
f(triple(a,b,c),A) |
(13) |
|
f''(triple(a,b,c)) |
→ |
foldC(triple(a,b,0),c) |
(15) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.