Termination w.r.t. Q proof of /tmp/tmpM4R1Tn/Ex4_7_77_Bor03

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:

a__zeros → cons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
a__tail(X) → tail(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

[a_{_zeros}, a_{_tail₁}, mark₁, tail₁] > cons₂
[a_{_zeros}, a_{_tail₁}, mark₁, tail₁] > 0
[a_{_zeros}, a_{_tail₁}, mark₁, tail₁] > zeros

Status:

cons₂: [1,2]
tail₁: multiset
a_{_tail₁}: multiset
a_{_zeros}: multiset
zeros: multiset
mark₁: multiset
0: multiset

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

a__zeros → cons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros

(2) Obligation:

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

mark(tail(X)) → a__tail(mark(X))
a__tail(X) → tail(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > a_{_tail₁} > tail₁

Status:

tail₁: multiset
a_{_tail₁}: multiset
mark₁: [1]

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

mark(tail(X)) → a__tail(mark(X))
a__tail(X) → tail(X)

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)