(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__FROM(X) → MARK(X)
A__HEAD(cons(X, XS)) → MARK(X)
A__2ND(cons(X, XS)) → A__HEAD(mark(XS))
A__2ND(cons(X, XS)) → MARK(XS)
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(head(X)) → A__HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(2nd(X)) → A__2ND(mark(X))
MARK(2nd(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.