R
↳Dependency Pair Analysis
ANATS -> AADX(azeros)
ANATS -> AZEROS
AADX(cons(X, Y)) -> AINCR(cons(X, adx(Y)))
AHD(cons(X, Y)) -> MARK(X)
ATL(cons(X, Y)) -> MARK(Y)
MARK(nats) -> ANATS
MARK(adx(X)) -> AADX(mark(X))
MARK(adx(X)) -> MARK(X)
MARK(zeros) -> AZEROS
MARK(incr(X)) -> AINCR(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(tl(X)) -> ATL(mark(X))
MARK(tl(X)) -> MARK(X)
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
MARK(tl(X)) -> MARK(X)
ATL(cons(X, Y)) -> MARK(Y)
MARK(tl(X)) -> ATL(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
AHD(cons(X, Y)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
MARK(tl(X)) -> MARK(X)
MARK(tl(X)) -> ATL(mark(X))
anats -> aadx(azeros)
anats -> nats
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
POL(a__nats) = 0 POL(MARK(x1)) = x1 POL(adx(x1)) = x1 POL(a__zeros) = 0 POL(incr(x1)) = x1 POL(A__TL(x1)) = x1 POL(a__hd(x1)) = x1 POL(a__tl(x1)) = 1 + x1 POL(tl(x1)) = 1 + x1 POL(mark(x1)) = x1 POL(a__adx(x1)) = x1 POL(A__HD(x1)) = x1 POL(0) = 0 POL(cons(x1, x2)) = x1 + x2 POL(hd(x1)) = x1 POL(nats) = 0 POL(s(x1)) = 0 POL(zeros) = 0 POL(a__incr(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Dependency Graph
ATL(cons(X, Y)) -> MARK(Y)
MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
AHD(cons(X, Y)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳DGraph
...
→DP Problem 3
↳Polynomial Ordering
MARK(hd(X)) -> MARK(X)
AHD(cons(X, Y)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
anats -> aadx(azeros)
anats -> nats
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
POL(a__nats) = 0 POL(MARK(x1)) = x1 POL(adx(x1)) = x1 POL(a__zeros) = 0 POL(incr(x1)) = x1 POL(a__hd(x1)) = 1 + x1 POL(a__tl(x1)) = x1 POL(tl(x1)) = x1 POL(mark(x1)) = x1 POL(a__adx(x1)) = x1 POL(A__HD(x1)) = x1 POL(0) = 0 POL(cons(x1, x2)) = x1 + x2 POL(hd(x1)) = 1 + x1 POL(nats) = 0 POL(s(x1)) = 0 POL(zeros) = 0 POL(a__incr(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳DGraph
...
→DP Problem 4
↳Dependency Graph
AHD(cons(X, Y)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳DGraph
...
→DP Problem 5
↳Polynomial Ordering
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
MARK(adx(X)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
POL(a__nats) = 1 POL(MARK(x1)) = 1 + x1 POL(adx(x1)) = 1 + x1 POL(a__zeros) = 0 POL(incr(x1)) = x1 POL(a__hd(x1)) = x1 POL(a__tl(x1)) = x1 POL(tl(x1)) = x1 POL(mark(x1)) = x1 POL(a__adx(x1)) = 1 + x1 POL(0) = 0 POL(cons(x1, x2)) = x1 + x2 POL(hd(x1)) = x1 POL(nats) = 1 POL(s(x1)) = 0 POL(zeros) = 0 POL(a__incr(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳DGraph
...
→DP Problem 6
↳Remaining Obligation(s)
MARK(incr(X)) -> MARK(X)
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)