(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0, cons(X, Z)) → mark(X)
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0, cons(X, Z)) → a__quote(X)
a__first1(0, Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0) → 01
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01) → 0
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01) → 01
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(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__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__FROM(X) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), mark(Z))
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
A__SEL1(0, cons(X, Z)) → A__QUOTE(X)
A__FIRST1(s(X), cons(Y, Z)) → A__QUOTE(Y)
A__FIRST1(s(X), cons(Y, Z)) → A__FIRST1(mark(X), mark(Z))
A__FIRST1(s(X), cons(Y, Z)) → MARK(X)
A__FIRST1(s(X), cons(Y, Z)) → MARK(Z)
A__QUOTE1(cons(X, Z)) → A__QUOTE(X)
A__QUOTE1(cons(X, Z)) → A__QUOTE1(Z)
A__QUOTE(s(X)) → A__QUOTE(X)
A__QUOTE(sel(X, Z)) → A__SEL1(mark(X), mark(Z))
A__QUOTE(sel(X, Z)) → MARK(X)
A__QUOTE(sel(X, Z)) → MARK(Z)
A__QUOTE1(first(X, Z)) → A__FIRST1(mark(X), mark(Z))
A__QUOTE1(first(X, Z)) → MARK(X)
A__QUOTE1(first(X, Z)) → MARK(Z)
A__UNQUOTE(s1(X)) → A__UNQUOTE(mark(X))
A__UNQUOTE(s1(X)) → MARK(X)
A__UNQUOTE1(cons1(X, Z)) → A__FCONS(a__unquote(mark(X)), a__unquote1(mark(Z)))
A__UNQUOTE1(cons1(X, Z)) → A__UNQUOTE(mark(X))
A__UNQUOTE1(cons1(X, Z)) → MARK(X)
A__UNQUOTE1(cons1(X, Z)) → A__UNQUOTE1(mark(Z))
A__UNQUOTE1(cons1(X, Z)) → MARK(Z)
A__FCONS(X, Z) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(quote(X)) → A__QUOTE(X)
MARK(first1(X1, X2)) → A__FIRST1(mark(X1), mark(X2))
MARK(first1(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(quote1(X)) → A__QUOTE1(X)
MARK(unquote(X)) → A__UNQUOTE(mark(X))
MARK(unquote(X)) → MARK(X)
MARK(unquote1(X)) → A__UNQUOTE1(mark(X))
MARK(unquote1(X)) → MARK(X)
MARK(fcons(X1, X2)) → A__FCONS(mark(X1), mark(X2))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(fcons(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(cons1(X1, X2)) → MARK(X1)
MARK(cons1(X1, X2)) → MARK(X2)
MARK(s1(X)) → MARK(X)
The TRS R consists of the following rules:
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel(0, cons(X, Z)) → mark(X)
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__sel1(0, cons(X, Z)) → a__quote(X)
a__first1(0, Z) → nil1
a__first1(s(X), cons(Y, Z)) → cons1(a__quote(Y), a__first1(mark(X), mark(Z)))
a__quote(0) → 01
a__quote1(cons(X, Z)) → cons1(a__quote(X), a__quote1(Z))
a__quote1(nil) → nil1
a__quote(s(X)) → s1(a__quote(X))
a__quote(sel(X, Z)) → a__sel1(mark(X), mark(Z))
a__quote1(first(X, Z)) → a__first1(mark(X), mark(Z))
a__unquote(01) → 0
a__unquote(s1(X)) → s(a__unquote(mark(X)))
a__unquote1(nil1) → nil
a__unquote1(cons1(X, Z)) → a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z)))
a__fcons(X, Z) → cons(mark(X), Z)
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(X)
mark(first1(X1, X2)) → a__first1(mark(X1), mark(X2))
mark(quote1(X)) → a__quote1(X)
mark(unquote(X)) → a__unquote(mark(X))
mark(unquote1(X)) → a__unquote1(mark(X))
mark(fcons(X1, X2)) → a__fcons(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(nil) → nil
mark(nil1) → nil1
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
mark(01) → 01
mark(s1(X)) → s1(mark(X))
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)
a__first1(X1, X2) → first1(X1, X2)
a__quote1(X) → quote1(X)
a__unquote(X) → unquote(X)
a__unquote1(X) → unquote1(X)
a__fcons(X1, X2) → fcons(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.