YES
proof of /TPDB/Applicative_05_ReverseLastInit.ari
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   app(app(app(compose, f), g), x) -> app(g, app(f, x))
   app(reverse, l) -> app(app(reverse2, l), nil)
   app(app(reverse2, nil), l) -> l
   app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
   app(hd, app(app(cons, x), xs)) -> x
   app(tl, app(app(cons, x), xs)) -> xs
   last -> app(app(compose, hd), reverse)
   init -> app(app(compose, reverse), app(app(compose, tl), reverse))

Q is empty.

----------------------------------------

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(app(x_1, x_2)) = x_1 + x_2
   POL(compose) = 0
   POL(cons) = 0
   POL(hd) = 1
   POL(init) = 2
   POL(last) = 2
   POL(nil) = 0
   POL(reverse) = 0
   POL(reverse2) = 0
   POL(tl) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   app(hd, app(app(cons, x), xs)) -> x
   last -> app(app(compose, hd), reverse)
   init -> app(app(compose, reverse), app(app(compose, tl), reverse))




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   app(app(app(compose, f), g), x) -> app(g, app(f, x))
   app(reverse, l) -> app(app(reverse2, l), nil)
   app(app(reverse2, nil), l) -> l
   app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
   app(tl, app(app(cons, x), xs)) -> xs

Q is empty.

----------------------------------------

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:tl > cons > nil > reverse2 > reverse > compose > app_2

and weight map:

   compose=1
   reverse=4
   reverse2=1
   nil=2
   cons=1
   tl=1
   app_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   app(app(app(compose, f), g), x) -> app(g, app(f, x))
   app(reverse, l) -> app(app(reverse2, l), nil)
   app(app(reverse2, nil), l) -> l
   app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
   app(tl, app(app(cons, x), xs)) -> xs




----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

----------------------------------------

(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(6)
YES