Certification Problem

Input (TPDB TRS_Standard/Strategy_removed_AG01/#4.33)

The rewrite relation of the following TRS is considered.

sum(cons(s(n),x),cons(m,y)) → sum(cons(n,x),cons(s(m),y)) (1)
sum(cons(0,x),y) → sum(x,y) (2)
sum(nil,y) → y (3)
weight(cons(n,cons(m,x))) → weight(sum(cons(n,cons(m,x)),cons(0,x))) (4)
weight(cons(n,nil)) → n (5)

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
[nil] =
0 0 0
1 0 0
0 0 0
[cons(x1, x2)] =
1 1 1
0 1 0
0 0 0
· x1 +
1 0 0
1 1 0
1 0 0
· x2 +
0 0 0
0 0 0
0 0 0
[0] =
0 0 0
0 0 0
0 0 0
[weight(x1)] =
1 1 0
0 1 0
1 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 1
0 0 0
0 1 0
· x1 +
1 0 0
0 0 0
1 0 0
[sum(x1, x2)] =
1 0 0
0 0 0
1 0 0
· x1 +
1 0 0
0 1 0
0 0 1
· x2 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
weight(cons(n,nil)) → n (5)

1.1 Rule Removal

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

1.1.1 Rule Removal

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

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[cons(x1, x2)] =
1 0 1
0 0 0
0 0 0
· x1 +
1 0 0
1 0 1
1 0 1
· x2 +
0 0 0
0 0 0
1 0 0
[0] =
0 0 0
0 0 0
0 0 0
[weight(x1)] =
1 1 0
0 0 0
1 1 0
· x1 +
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
[sum(x1, x2)] =
1 0 0
0 0 0
1 0 0
· x1 +
1 1 0
0 0 0
0 1 0
· x2 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
weight(cons(n,cons(m,x))) → weight(sum(cons(n,cons(m,x)),cons(0,x))) (4)

1.1.1.1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(sum) = 0 weight(sum) = 0
prec(cons) = 2 weight(cons) = 0
prec(s) = 3 weight(s) = 2
all of the following rules can be deleted.
sum(cons(s(n),x),cons(m,y)) → sum(cons(n,x),cons(s(m),y)) (1)

1.1.1.1.1.1 R is empty

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