(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
terms(
x1) =
terms(
x1)
cons(
x1,
x2) =
cons(
x1,
x2)
recip(
x1) =
x1
sqr(
x1) =
sqr(
x1)
n__terms(
x1) =
x1
s(
x1) =
s(
x1)
0 =
0
add(
x1,
x2) =
add(
x1,
x2)
dbl(
x1) =
dbl(
x1)
first(
x1,
x2) =
first(
x1,
x2)
nil =
nil
n__first(
x1,
x2) =
n__first(
x1,
x2)
activate(
x1) =
activate(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
0 > nil > s1
[first2, nfirst2] > activate1 > terms1 > cons2 > s1
[first2, nfirst2] > activate1 > terms1 > sqr1 > add2 > s1
[first2, nfirst2] > activate1 > terms1 > sqr1 > dbl1 > s1
Status:
terms1: [1]
cons2: [2,1]
sqr1: multiset
s1: multiset
0: multiset
add2: [1,2]
dbl1: multiset
first2: [1,2]
nil: multiset
nfirst2: [1,2]
activate1: [1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
first(X1, X2) → n__first(X1, X2)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
first2 > nfirst2
and weight map:
first_2=0
n__first_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:
first(X1, X2) → n__first(X1, X2)
(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