(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:
Combined order from the following AFS and order.
a__eq(
x1,
x2) =
a__eq(
x1,
x2)
0 =
0
true =
true
s(
x1) =
x1
false =
false
a__inf(
x1) =
a__inf(
x1)
cons(
x1,
x2) =
cons(
x1,
x2)
inf(
x1) =
inf(
x1)
a__take(
x1,
x2) =
a__take(
x1,
x2)
nil =
nil
take(
x1,
x2) =
take(
x1,
x2)
a__length(
x1) =
x1
length(
x1) =
x1
mark(
x1) =
mark(
x1)
eq(
x1,
x2) =
eq(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > aeq2 > false
mark1 > aeq2 > eq2
mark1 > [0, atake2, nil, take2] > true
mark1 > [0, atake2, nil, take2] > cons2
mark1 > ainf1 > cons2
mark1 > ainf1 > inf1
Status:
aeq2: multiset
0: multiset
true: multiset
false: multiset
ainf1: [1]
cons2: [2,1]
inf1: [1]
atake2: multiset
nil: multiset
take2: multiset
mark1: multiset
eq2: multiset
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(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(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(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)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__length(nil) → 0
mark(length(X)) → a__length(mark(X))
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
s1 > mark1 > take2 > atake2 > alength1 > aeq2 > length1 > 0 > nil
and weight map:
nil=1
0=2
s_1=0
a__length_1=1
mark_1=2
length_1=1
a__take_2=1
take_2=0
a__eq_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:
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__length(nil) → 0
mark(length(X)) → a__length(mark(X))
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES