Certification Problem

Input (TPDB TRS_Relative/INVY_15/#3.17_mset)

The relative rewrite relation R/S is considered where R is the following TRS

app(nil,k) → k (1)
app(l,nil) → l (2)
app(cons(x,l),k) → cons(x,app(l,k)) (3)
sum(cons(x,nil)) → cons(x,nil) (4)
sum(cons(x,cons(y,l))) → sum(cons(plus(x,y),l)) (5)
sum(app(l,cons(x,cons(y,k)))) → sum(app(l,sum(cons(x,cons(y,k))))) (6)
plus(0,y) → y (7)
plus(s(x),y) → s(plus(x,y)) (8)

and S is the following TRS.

cons(x,cons(y,l)) → cons(y,cons(x,l)) (9)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[plus(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
1 1 0
0 1 1
· x2 +
0 0 0
0 0 0
0 0 0
[app(x1, x2)] =
1 0 0
0 1 0
0 0 1
· x1 +
1 0 0
1 1 1
1 0 1
· x2 +
0 0 0
0 0 0
0 0 0
[sum(x1)] =
1 0 0
1 0 0
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[0] =
1 0 0
0 0 0
0 0 0
[nil] =
0 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[cons(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
· x2 +
0 0 0
0 0 0
1 0 0
all of the following rules can be deleted.
plus(0,y) → y (7)

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[plus(x1, x2)] =
1 0 0
0 0 0
0 0 1
· x1 +
1 0 0
0 0 0
0 0 1
· x2 +
0 0 0
0 0 0
0 0 0
[app(x1, x2)] =
1 0 0
1 1 0
0 0 1
· x1 +
1 1 0
0 1 0
0 0 1
· x2 +
1 0 0
0 0 0
0 0 0
[sum(x1)] =
1 0 0
0 1 0
1 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[nil] =
0 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[cons(x1, x2)] =
1 0 1
0 0 1
0 0 0
· x1 +
1 0 0
0 1 0
0 0 0
· x2 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
app(nil,k) → k (1)
app(l,nil) → l (2)

1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[plus(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 1
0 1 0
· x2 +
1 0 0
0 0 0
0 0 0
[app(x1, x2)] =
1 1 0
1 0 0
0 0 0
· x1 +
1 0 0
0 0 1
0 0 0
· x2 +
0 0 0
0 0 0
1 0 0
[sum(x1)] =
1 0 0
0 0 0
1 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[nil] =
1 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
[cons(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 1 0
0 0 0
· x2 +
1 0 0
1 0 0
0 0 0
all of the following rules can be deleted.
app(cons(x,l),k) → cons(x,app(l,k)) (3)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[plus(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
· x2 +
1 0 0
0 0 0
0 0 0
[app(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
· x2 +
0 0 0
0 0 0
0 0 0
[sum(x1)] =
1 0 0
0 1 0
1 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[nil] =
0 0 0
0 0 0
1 0 0
[s(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[cons(x1, x2)] =
1 0 0
1 0 0
0 0 0
· x1 +
1 0 1
0 1 0
0 0 1
· x2 +
1 0 0
1 0 0
1 0 0
all of the following rules can be deleted.
sum(cons(x,cons(y,l))) → sum(cons(plus(x,y),l)) (5)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[plus(x1, x2)] =
1 0 1
1 0 0
1 0 1
· x1 +
1 1 1
0 0 1
1 0 0
· x2 +
0 0 0
0 0 0
1 0 0
[app(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
· x2 +
0 0 0
0 0 0
1 0 0
[sum(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[nil] =
0 0 0
1 0 0
0 0 0
[s(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
1 0 0
0 0 0
1 0 0
[cons(x1, x2)] =
1 1 0
0 0 0
0 0 0
· x1 +
1 1 0
0 0 0
0 0 0
· x2 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
plus(s(x),y) → s(plus(x,y)) (8)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[app(x1, x2)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 1 0
0 0 0
1 0 0
· x2 +
0 0 0
1 0 0
0 0 0
[sum(x1)] =
1 0 0
0 0 0
0 1 0
· x1 +
0 0 0
1 0 0
1 0 0
[nil] =
0 0 0
0 0 0
0 0 0
[cons(x1, x2)] =
1 0 1
0 0 0
0 0 0
· x1 +
1 1 0
0 1 0
0 0 0
· x2 +
0 0 0
1 0 0
0 0 0
all of the following rules can be deleted.
sum(app(l,cons(x,cons(y,k)))) → sum(app(l,sum(cons(x,cons(y,k))))) (6)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals
[sum(x1)] =
1 0 0 0
0 1 0 0
0 0 1 0
0 1 0 0
· x1 +
1 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
[nil] =
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
[cons(x1, x2)] =
1 0 0 0
0 0 1 0
0 0 1 0
0 0 1 0
· x1 +
1 0 0 0
0 0 1 0
0 0 0 1
0 0 1 0
· x2 +
0 0 0 0
0 0 0 0
1 0 0 0
1 0 0 0
all of the following rules can be deleted.
sum(cons(x,nil)) → cons(x,nil) (4)

1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.