Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex8_BLR02_FR)

The rewrite relation of the following TRS is considered.

fib(N) → sel(N,fib1(s(0),s(0))) (1)
fib1(X,Y) → cons(X,n__fib1(Y,n__add(X,Y))) (2)
add(0,X) → X (3)
add(s(X),Y) → s(add(X,Y)) (4)
sel(0,cons(X,XS)) → X (5)
sel(s(N),cons(X,XS)) → sel(N,activate(XS)) (6)
fib1(X1,X2) → n__fib1(X1,X2) (7)
add(X1,X2) → n__add(X1,X2) (8)
activate(n__fib1(X1,X2)) → fib1(activate(X1),activate(X2)) (9)
activate(n__add(X1,X2)) → add(activate(X1),activate(X2)) (10)
activate(X) → X (11)

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(activate) = 3 status(activate) = [1] list-extension(activate) = Lex
prec(add) = 2 status(add) = [1, 2] list-extension(add) = Lex
prec(cons) = 0 status(cons) = [2, 1] list-extension(cons) = Lex
prec(n__fib1) = 0 status(n__fib1) = [2, 1] list-extension(n__fib1) = Lex
prec(n__add) = 0 status(n__add) = [2, 1] list-extension(n__add) = Lex
prec(sel) = 4 status(sel) = [1, 2] list-extension(sel) = Lex
prec(fib1) = 2 status(fib1) = [2, 1] list-extension(fib1) = Lex
prec(s) = 0 status(s) = [1] list-extension(s) = Lex
prec(0) = 0 status(0) = [] list-extension(0) = Lex
prec(fib) = 7 status(fib) = [1] list-extension(fib) = Lex
and the following Max-polynomial interpretation
[activate(x1)] = 0 + 1 · x1
[add(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[cons(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[n__fib1(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[n__add(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[sel(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[fib1(x1, x2)] = max(0, 0 + 1 · x1, 0 + 1 · x2)
[s(x1)] = 0 + 1 · x1
[0] = max(0)
[fib(x1)] = max(1, 0 + 1 · x1)
all of the following rules can be deleted.
fib(N) → sel(N,fib1(s(0),s(0))) (1)
fib1(X,Y) → cons(X,n__fib1(Y,n__add(X,Y))) (2)
add(0,X) → X (3)
add(s(X),Y) → s(add(X,Y)) (4)
sel(0,cons(X,XS)) → X (5)
sel(s(N),cons(X,XS)) → sel(N,activate(XS)) (6)
fib1(X1,X2) → n__fib1(X1,X2) (7)
add(X1,X2) → n__add(X1,X2) (8)
activate(n__fib1(X1,X2)) → fib1(activate(X1),activate(X2)) (9)
activate(n__add(X1,X2)) → add(activate(X1),activate(X2)) (10)
activate(X) → X (11)

1.1 R is empty

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