Certification Problem
Input (TPDB TRS_Standard/CiME_04/list-sum-prod-assoc)
The rewrite relation of the following TRS is considered.
|
+(x,0) |
→ |
x |
(1) |
|
+(0,x) |
→ |
x |
(2) |
|
+(s(x),s(y)) |
→ |
s(s(+(x,y))) |
(3) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(4) |
|
*(x,0) |
→ |
0 |
(5) |
|
*(0,x) |
→ |
0 |
(6) |
|
*(s(x),s(y)) |
→ |
s(+(*(x,y),+(x,y))) |
(7) |
|
*(*(x,y),z) |
→ |
*(x,*(y,z)) |
(8) |
|
sum(nil) |
→ |
0 |
(9) |
|
sum(cons(x,l)) |
→ |
+(x,sum(l)) |
(10) |
|
prod(nil) |
→ |
s(0) |
(11) |
|
prod(cons(x,l)) |
→ |
*(x,prod(l)) |
(12) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
Weighted Path Order with the following precedence and status
| prec(prod) |
= |
0 |
|
status(prod) |
= |
[1] |
|
list-extension(prod) |
= |
Lex |
| prec(cons) |
= |
0 |
|
status(cons) |
= |
[2, 1] |
|
list-extension(cons) |
= |
Lex |
| prec(sum) |
= |
1 |
|
status(sum) |
= |
[1] |
|
list-extension(sum) |
= |
Lex |
| prec(nil) |
= |
0 |
|
status(nil) |
= |
[] |
|
list-extension(nil) |
= |
Lex |
| prec(*) |
= |
3 |
|
status(*) |
= |
[1, 2] |
|
list-extension(*) |
= |
Lex |
| prec(s) |
= |
0 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
| prec(+) |
= |
1 |
|
status(+) |
= |
[1, 2] |
|
list-extension(+) |
= |
Lex |
| prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
and the following
Max-polynomial interpretation
| [prod(x1)] |
=
|
4 + 1 · x1
|
| [cons(x1, x2)] |
=
|
1 + 1 · x1 + 1 · x2
|
| [sum(x1)] |
=
|
max(0, 2 + 1 · x1) |
| [nil] |
=
|
max(4) |
| [*(x1, x2)] |
=
|
max(2, 2 + 1 · x1, 0 + 1 · x2) |
| [s(x1)] |
=
|
max(0, 0 + 1 · x1) |
| [+(x1, x2)] |
=
|
max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [0] |
=
|
max(0) |
all of the following rules can be deleted.
|
+(x,0) |
→ |
x |
(1) |
|
+(0,x) |
→ |
x |
(2) |
|
+(s(x),s(y)) |
→ |
s(s(+(x,y))) |
(3) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(4) |
|
*(x,0) |
→ |
0 |
(5) |
|
*(0,x) |
→ |
0 |
(6) |
|
*(s(x),s(y)) |
→ |
s(+(*(x,y),+(x,y))) |
(7) |
|
*(*(x,y),z) |
→ |
*(x,*(y,z)) |
(8) |
|
sum(nil) |
→ |
0 |
(9) |
|
sum(cons(x,l)) |
→ |
+(x,sum(l)) |
(10) |
|
prod(nil) |
→ |
s(0) |
(11) |
|
prod(cons(x,l)) |
→ |
*(x,prod(l)) |
(12) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.