Termination w.r.t. Q proof of TRS/SK902.45.trs

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:

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

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

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y

Q is empty.

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

Lexicographic path order with status [19].
Precedence:

admit₂ > nil > cond₂
admit₂ > .₂ > cond₂
admit₂ > w > cond₂
admit₂ > =₂ > cond₂
admit₂ > sum₃ > cond₂
admit₂ > carry₃ > cond₂
true > cond₂

Status:

sum₃: multiset
true: multiset
w: multiset
admit₂: [2,1]
carry₃: multiset
=₂: multiset
nil: multiset
.₂: [1,2]
cond₂: multiset