Termination w.r.t. Q proof of /tmp/tmpzxKLO5/Ex1_GL02a

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__eq(0, 0) → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:

[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_inf₁} > cons₂ > [0, a_{_length₁}, length₁] > nil
[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_inf₁} > cons₂ > s₁
[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_inf₁} > cons₂ > take₂
[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_inf₁} > inf₁
[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_take₂} > cons₂ > [0, a_{_length₁}, length₁] > nil
[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_take₂} > cons₂ > s₁
[a_{_eq₂}, false, eq₂] > [true, mark₁] > a_{_take₂} > cons₂ > take₂

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

a__eq(0, 0) → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)

(2) Obligation:

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

a__eq(X1, X2) → eq(X1, X2)
a__length(X) → length(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:

a_{_eq₂} > [eq₂, length₁]
a_{_length₁} > [eq₂, length₁]

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

a__eq(X1, X2) → eq(X1, X2)
a__length(X) → length(X)

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)