(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
natsFrom(
x1) =
natsFrom(
x1)
cons(
x1,
x2) =
cons(
x1,
x2)
n__natsFrom(
x1) =
n__natsFrom(
x1)
n__s(
x1) =
n__s(
x1)
fst(
x1) =
x1
pair(
x1,
x2) =
pair(
x1,
x2)
snd(
x1) =
x1
splitAt(
x1,
x2) =
splitAt(
x1,
x2)
0 =
0
nil =
nil
s(
x1) =
s(
x1)
u(
x1,
x2,
x3,
x4) =
u(
x1,
x2,
x3,
x4)
activate(
x1) =
activate(
x1)
head(
x1) =
x1
tail(
x1) =
tail(
x1)
sel(
x1,
x2) =
sel(
x1,
x2)
afterNth(
x1,
x2) =
afterNth(
x1,
x2)
take(
x1,
x2) =
take(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[splitAt2, u4, sel2, afterNth2, take2] > [0, nil] > pair2
[splitAt2, u4, sel2, afterNth2, take2] > activate1 > natsFrom1 > cons2 > pair2
[splitAt2, u4, sel2, afterNth2, take2] > activate1 > natsFrom1 > nnatsFrom1 > pair2
[splitAt2, u4, sel2, afterNth2, take2] > activate1 > natsFrom1 > ns1 > pair2
[splitAt2, u4, sel2, afterNth2, take2] > activate1 > s1 > ns1 > pair2
tail1 > activate1 > natsFrom1 > cons2 > pair2
tail1 > activate1 > natsFrom1 > nnatsFrom1 > pair2
tail1 > activate1 > natsFrom1 > ns1 > pair2
tail1 > activate1 > s1 > ns1 > pair2
Status:
natsFrom1: [1]
cons2: [1,2]
nnatsFrom1: multiset
ns1: multiset
pair2: multiset
splitAt2: [1,2]
0: multiset
nil: multiset
s1: multiset
u4: [2,4,3,1]
activate1: [1]
tail1: multiset
sel2: [1,2]
afterNth2: [1,2]
take2: [1,2]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
afterNth2 > snd1 > splitAt2 > take2 > fst1 > sel2 > head1
and weight map:
head_1=1
fst_1=1
snd_1=1
sel_2=2
afterNth_2=1
take_2=1
splitAt_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:
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
(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