Termination w.r.t. Q proof of TRS/TRCSREx2_Luc03b

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

fst(0, Z) → nil
fst(s, cons(Y)) → cons(Y)
from(X) → cons(X)
add(0, X) → X
add(s, Y) → s
len(nil) → 0
len(cons(X)) → s

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

Q restricted rewrite system:
The TRS R consists of the following rules:

fst(0, Z) → nil
fst(s, cons(Y)) → cons(Y)
from(X) → cons(X)
add(0, X) → X
add(s, Y) → s
len(nil) → 0
len(cons(X)) → s

Q is empty.

We use [23] with the following order to prove termination.

Recursive path order with status [2].
Quasi-Precedence:

from₁ > [s, cons_1] > [fst₂, nil]
add₂ > [s, cons_1] > [fst₂, nil]
len₁ > 0 > [fst₂, nil]
len₁ > [s, cons_1] > [fst₂, nil]

Status:

from₁: multiset
len₁: multiset
fst₂: multiset
add₂: multiset
0: multiset
s: multiset
cons₁: multiset
nil: multiset