Termination w.r.t. Q proof of /tmp/tmpF5TGk7/ExIntrod_GM04

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__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
a__hd(cons(X, Y)) → mark(X)
a__tl(cons(X, Y)) → mark(Y)
mark(nats) → a__nats
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
mark(hd(X)) → a__hd(mark(X))
mark(tl(X)) → a__tl(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__nats → nats
a__adx(X) → adx(X)
a__zeros → zeros
a__incr(X) → incr(X)
a__hd(X) → hd(X)
a__tl(X) → tl(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

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

[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_adx₁} > a_{_incr₁} > cons₂
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_adx₁} > a_{_incr₁} > s₁
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_adx₁} > a_{_incr₁} > incr₁
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_adx₁} > adx₁
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_zeros} > cons₂
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_zeros} > 0
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > a_{_zeros} > zeros
[a_{_hd₁}, mark₁, a_{_tl₁}, hd₁, tl₁] > a_{_nats} > nats

Status:

trivial

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

a__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
a__hd(cons(X, Y)) → mark(X)
a__tl(cons(X, Y)) → mark(Y)
mark(nats) → a__nats
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__nats → nats
a__adx(X) → adx(X)
a__zeros → zeros
a__incr(X) → incr(X)

(2) Obligation:

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

mark(hd(X)) → a__hd(mark(X))
mark(tl(X)) → a__tl(mark(X))
a__hd(X) → hd(X)
a__tl(X) → tl(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

mark₁ > a_{_hd₁} > hd₁
mark₁ > a_{_tl₁} > tl₁