Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex7_BLR02_Z)

The rewrite relation of the following TRS is considered.

from(X) → cons(X,n__from(s(X))) (1)
head(cons(X,XS)) → X (2)
2nd(cons(X,XS)) → head(activate(XS)) (3)
take(0,XS) → nil (4)
take(s(N),cons(X,XS)) → cons(X,n__take(N,activate(XS))) (5)
sel(0,cons(X,XS)) → X (6)
sel(s(N),cons(X,XS)) → sel(N,activate(XS)) (7)
from(X) → n__from(X) (8)
take(X1,X2) → n__take(X1,X2) (9)
activate(n__from(X)) → from(X) (10)
activate(n__take(X1,X2)) → take(X1,X2) (11)
activate(X) → X (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(sel) = 12 status(sel) = [1, 2] list-extension(sel) = Lex
prec(n__take) = 0 status(n__take) = [2, 1] list-extension(n__take) = Lex
prec(nil) = 0 status(nil) = [] list-extension(nil) = Lex
prec(take) = 1 status(take) = [2, 1] list-extension(take) = Lex
prec(0) = 0 status(0) = [] list-extension(0) = Lex
prec(activate) = 8 status(activate) = [1] list-extension(activate) = Lex
prec(2nd) = 0 status(2nd) = [1] list-extension(2nd) = Lex
prec(head) = 0 status(head) = [1] list-extension(head) = Lex
prec(cons) = 0 status(cons) = [1, 2] list-extension(cons) = Lex
prec(n__from) = 0 status(n__from) = [1] list-extension(n__from) = Lex
prec(s) = 0 status(s) = [1] list-extension(s) = Lex
prec(from) = 1 status(from) = [1] list-extension(from) = Lex
and the following Max-polynomial interpretation
[sel(x1, x2)] = max(0, 2 + 1 · x1, 0 + 1 · x2)
[n__take(x1, x2)] = max(0, 4 + 1 · x1, 2 + 1 · x2)
[nil] = max(0)
[take(x1, x2)] = max(0, 4 + 1 · x1, 2 + 1 · x2)
[0] = max(4)
[activate(x1)] = max(0, 0 + 1 · x1)
[2nd(x1)] = 4 + 1 · x1
[head(x1)] = max(1, 0 + 1 · x1)
[cons(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[n__from(x1)] = max(0, 0 + 1 · x1)
[s(x1)] = 0 + 1 · x1
[from(x1)] = max(0, 0 + 1 · x1)
all of the following rules can be deleted.
from(X) → cons(X,n__from(s(X))) (1)
head(cons(X,XS)) → X (2)
2nd(cons(X,XS)) → head(activate(XS)) (3)
take(0,XS) → nil (4)
take(s(N),cons(X,XS)) → cons(X,n__take(N,activate(XS))) (5)
sel(0,cons(X,XS)) → X (6)
sel(s(N),cons(X,XS)) → sel(N,activate(XS)) (7)
from(X) → n__from(X) (8)
take(X1,X2) → n__take(X1,X2) (9)
activate(n__from(X)) → from(X) (10)
activate(n__take(X1,X2)) → take(X1,X2) (11)
activate(X) → X (12)

1.1 R is empty

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