0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 QDPSizeChangeProof (⇔)
↳4 TRUE
a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
A__AND(tt, X) → MARK(X)
A__PLUS(N, 0) → MARK(N)
A__PLUS(N, s(M)) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → MARK(N)
A__PLUS(N, s(M)) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
Order:Combined order from the following AFS and order.
mark(x1) = x1
and(x1, x2) = and(x1, x2)
a__and(x1, x2) = a__and(x1, x2)
tt = tt
plus(x1, x2) = plus(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
s(x1) = s(x1)
Recursive path order with status [RPO].
Quasi-Precedence:
[and2, aand2] > s1
tt > s1
[plus2, aplus2] > s1
0 > s1
and2: [2,1]
aand2: [2,1]
tt: multiset
plus2: [2,1]
aplus2: [2,1]
0: multiset
s1: multiset
AFS:
mark(x1) = x1
and(x1, x2) = and(x1, x2)
a__and(x1, x2) = a__and(x1, x2)
tt = tt
plus(x1, x2) = plus(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
We oriented the following set of usable rules [AAECC05,FROCOS05].
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)