Termination w.r.t. Q proof of /tmp/tmpqAHKSH/Ex3_12_Luc96a

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 RisEmptyProof (⇔)

↳6 TRUE

(0) Obligation:

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

from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

sel₂ > activate₁ > from₁ > cons₂ > [n_{_s₁}, s₁]
sel₂ > activate₁ > from₁ > n_{_from₁} > [n_{_s₁}, s₁]
0 > [n_{_s₁}, s₁]

Status:

sel₂: [1,2]
cons₂: [2,1]
from₁: [1]
n_{_from₁}: [1]
s₁: [1]
activate₁: [1]
n_{_s₁}: [1]
0: []

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

(2) Obligation:

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

s(X) → n__s(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

s₁ > n_{_s₁}

Status:

s₁: [1]
n_{_s₁}: [1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

s(X) → n__s(X)

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)