(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
natsFrom(N) → cons(N, n__natsFrom(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
tail1 > [natsFrom1, u4, activate1] > s1 > nnatsFrom1
tail1 > [natsFrom1, u4, activate1] > [pair2, 0, nil] > cons2 > nnatsFrom1
sel2 > head1 > nnatsFrom1
sel2 > afterNth2 > snd1 > nnatsFrom1
sel2 > afterNth2 > splitAt2 > [natsFrom1, u4, activate1] > s1 > nnatsFrom1
sel2 > afterNth2 > splitAt2 > [natsFrom1, u4, activate1] > [pair2, 0, nil] > cons2 > nnatsFrom1
take2 > fst1 > nnatsFrom1
take2 > splitAt2 > [natsFrom1, u4, activate1] > s1 > nnatsFrom1
take2 > splitAt2 > [natsFrom1, u4, activate1] > [pair2, 0, nil] > cons2 > nnatsFrom1
Status:
sel2: [1,2]
tail1: multiset
afterNth2: multiset
snd1: multiset
head1: multiset
nnatsFrom1: multiset
activate1: [1]
take2: multiset
splitAt2: [1,2]
0: multiset
cons2: multiset
u4: [3,2,1,4]
fst1: multiset
pair2: [1,2]
s1: multiset
natsFrom1: [1]
nil: multiset
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(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) TRUE