(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__nats → cons(0, incr(nats))
a__pairs → cons(0, incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)
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__ODDS → A__INCR(a__pairs)
A__ODDS → A__PAIRS
A__INCR(cons(X, XS)) → MARK(X)
A__HEAD(cons(X, XS)) → MARK(X)
A__TAIL(cons(X, XS)) → MARK(XS)
MARK(nats) → A__NATS
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(pairs) → A__PAIRS
MARK(odds) → A__ODDS
MARK(head(X)) → A__HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → A__TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__nats → cons(0, incr(nats))
a__pairs → cons(0, incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__INCR(cons(X, XS)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(odds) → A__ODDS
A__ODDS → A__INCR(a__pairs)
MARK(head(X)) → A__HEAD(mark(X))
A__HEAD(cons(X, XS)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → A__TAIL(mark(X))
A__TAIL(cons(X, XS)) → MARK(XS)
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__nats → cons(0, incr(nats))
a__pairs → cons(0, incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.