TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60000 ms.

The following DP Processors were used

Problem 1 was processed with processor DependencyGraph (20944ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 24 remains open; application of the following processors failed [DependencyGraph (16ms), PolynomialLinearRange4iUR (5ms)].
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 25 remains open; application of the following processors failed [DependencyGraph (6ms), PolynomialLinearRange4iUR (10ms)].
 | – Problem 4 was processed with processor DependencyGraph (3984ms).
 |    | – Problem 30 remains open; application of the following processors failed [PolynomialLinearRange4iUR (8ms)].
 | – Problem 5 was processed with processor SubtermCriterion (4ms).
 | – Problem 6 was processed with processor DependencyGraph (4106ms).
 |    | – Problem 31 remains open; application of the following processors failed [PolynomialLinearRange4iUR (7ms)].
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 | – Problem 8 was processed with processor SubtermCriterion (1ms).
 | – Problem 9 was processed with processor DependencyGraph (3983ms).
 |    | – Problem 32 remains open; application of the following processors failed [PolynomialLinearRange4iUR (25ms), DependencyGraph (3871ms)].
 | – Problem 10 was processed with processor SubtermCriterion (1ms).
 | – Problem 11 was processed with processor DependencyGraph (3991ms).
 |    | – Problem 33 remains open; application of the following processors failed [PolynomialLinearRange4iUR (23ms), DependencyGraph (timeout)].
 | – Problem 12 was processed with processor SubtermCriterion (3ms).
 | – Problem 13 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (3827ms), PolynomialLinearRange4iUR (7ms)].
 | – Problem 14 was processed with processor DependencyGraph (3975ms).
 |    | – Problem 34 remains open; application of the following processors failed [PolynomialLinearRange4iUR (833ms), DependencyGraph (3964ms)].
 | – Problem 15 was processed with processor SubtermCriterion (1ms).
 | – Problem 16 was processed with processor SubtermCriterion (2ms).
 | – Problem 17 was processed with processor DependencyGraph (4098ms).
 |    | – Problem 29 remains open; application of the following processors failed [PolynomialLinearRange4iUR (8ms)].
 | – Problem 18 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 26 remains open; application of the following processors failed [DependencyGraph (6ms), PolynomialLinearRange4iUR (7ms)].
 | – Problem 19 was processed with processor SubtermCriterion (1ms).
 | – Problem 20 was processed with processor SubtermCriterion (1ms).
 | – Problem 21 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 27 remains open; application of the following processors failed [DependencyGraph (6ms), PolynomialLinearRange4iUR (1ms)].
 | – Problem 22 was processed with processor SubtermCriterion (1ms).
 | – Problem 23 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 28 remains open; application of the following processors failed [DependencyGraph (44ms), PolynomialLinearRange4iUR (5ms)].

The following open problems remain:

Open Dependency Pair Problem 34

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(length(X))	→	active^#(length(mark(X)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Open Dependency Pair Problem 32

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(length(X))	→	active^#(length(mark(X)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Open Dependency Pair Problem 33

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(length(X))	→	active^#(length(mark(X)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Open Dependency Pair Problem 25

Dependency Pairs

cons^#(X1, active(X2))

→

cons^#(X1, X2)

cons^#(X1, mark(X2))

→

cons^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Open Dependency Pair Problem 24

Dependency Pairs

U61^#(X1, X2, active(X3))	→	U61^#(X1, X2, X3)		U61^#(X1, X2, mark(X3))	→	U61^#(X1, X2, X3)
U61^#(X1, mark(X2), X3)	→	U61^#(X1, X2, X3)		U61^#(X1, active(X2), X3)	→	U61^#(X1, X2, X3)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Open Dependency Pair Problem 27

Dependency Pairs

U62^#(X1, active(X2))

→

U62^#(X1, X2)

U62^#(X1, mark(X2))

→

U62^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Open Dependency Pair Problem 26

Dependency Pairs

U51^#(X1, mark(X2))

→

U51^#(X1, X2)

U51^#(X1, active(X2))

→

U51^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Open Dependency Pair Problem 29

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(length(X))	→	active^#(length(mark(X)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Open Dependency Pair Problem 28

Dependency Pairs

U41^#(X1, mark(X2))

→

U41^#(X1, X2)

U41^#(X1, active(X2))

→

U41^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Open Dependency Pair Problem 13

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U62(X1, X2))	→	mark^#(X1)
mark^#(U42(X))	→	mark^#(X)	mark^#(U51(X1, X2))	→	mark^#(X1)
mark^#(U21(X))	→	mark^#(X)	active^#(U11(tt))	→	mark^#(tt)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(U41(X1, X2))	→	mark^#(X1)	mark^#(U11(X))	→	active^#(U11(mark(X)))
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(isNatList(X))	→	active^#(isNatList(X))	mark^#(length(X))	→	active^#(length(mark(X)))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Open Dependency Pair Problem 31

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(length(X))	→	active^#(length(mark(X)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Open Dependency Pair Problem 30

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(length(X))	→	mark^#(X)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
mark^#(s(X))	→	active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(length(X))	→	active^#(length(mark(X)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, U31, U21, nil, cons

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

U62^#(X1, active(X2))	→	U62^#(X1, X2)	mark^#(U51(X1, X2))	→	U51^#(mark(X1), X2)
active^#(U41(tt, V2))	→	isNatIList^#(V2)	active^#(length(nil))	→	mark^#(0)
U62^#(active(X1), X2)	→	U62^#(X1, X2)	active^#(U51(tt, V2))	→	U52^#(isNatList(V2))
active^#(U51(tt, V2))	→	isNatList^#(V2)	U61^#(X1, active(X2), X3)	→	U61^#(X1, X2, X3)
U61^#(X1, X2, mark(X3))	→	U61^#(X1, X2, X3)	U51^#(mark(X1), X2)	→	U51^#(X1, X2)
active^#(isNat(length(V1)))	→	isNatList^#(V1)	isNat^#(active(X))	→	isNat^#(X)
mark^#(s(X))	→	mark^#(X)	mark^#(U42(X))	→	U42^#(mark(X))
active^#(U62(tt, L))	→	length^#(L)	length^#(mark(X))	→	length^#(X)
U62^#(X1, mark(X2))	→	U62^#(X1, X2)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
U41^#(active(X1), X2)	→	U41^#(X1, X2)	active^#(isNatIList(cons(V1, V2)))	→	isNat^#(V1)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	mark^#(U31(X))	→	mark^#(X)
active^#(U61(tt, L, N))	→	U62^#(isNat(N), L)	active^#(U42(tt))	→	mark^#(tt)
mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))	U51^#(X1, active(X2))	→	U51^#(X1, X2)
U61^#(active(X1), X2, X3)	→	U61^#(X1, X2, X3)	cons^#(mark(X1), X2)	→	cons^#(X1, X2)
mark^#(tt)	→	active^#(tt)	mark^#(U11(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNat(length(V1)))	→	U11^#(isNatList(V1))
active^#(U41(tt, V2))	→	U42^#(isNatIList(V2))	mark^#(length(X))	→	mark^#(X)
active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))	cons^#(X1, mark(X2))	→	cons^#(X1, X2)
mark^#(U41(X1, X2))	→	mark^#(X1)	mark^#(U11(X))	→	active^#(U11(mark(X)))
U41^#(X1, active(X2))	→	U41^#(X1, X2)	mark^#(zeros)	→	active^#(zeros)
U61^#(mark(X1), X2, X3)	→	U61^#(X1, X2, X3)	active^#(isNat(s(V1)))	→	isNat^#(V1)
U11^#(mark(X))	→	U11^#(X)	mark^#(0)	→	active^#(0)
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(s(X))	→	active^#(s(mark(X)))
active^#(isNat(s(V1)))	→	U21^#(isNat(V1))	mark^#(U61(X1, X2, X3))	→	U61^#(mark(X1), X2, X3)
cons^#(active(X1), X2)	→	cons^#(X1, X2)	active^#(isNatList(cons(V1, V2)))	→	U51^#(isNat(V1), V2)
U51^#(X1, mark(X2))	→	U51^#(X1, X2)	isNatList^#(mark(X))	→	isNatList^#(X)
mark^#(isNatList(X))	→	isNatList^#(X)	active^#(zeros)	→	cons^#(0, zeros)
mark^#(U31(X))	→	U31^#(mark(X))	active^#(isNatIList(V))	→	U31^#(isNatList(V))
mark^#(U21(X))	→	U21^#(mark(X))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))
mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
U11^#(active(X))	→	U11^#(X)	active^#(U21(tt))	→	mark^#(tt)
active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))	active^#(isNatIList(cons(V1, V2)))	→	U41^#(isNat(V1), V2)
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	U42^#(active(X))	→	U42^#(X)
mark^#(s(X))	→	s^#(mark(X))	mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))
U42^#(mark(X))	→	U42^#(X)	mark^#(U52(X))	→	U52^#(mark(X))
length^#(active(X))	→	length^#(X)	U52^#(active(X))	→	U52^#(X)
active^#(length(cons(N, L)))	→	U61^#(isNatList(L), L, N)	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
U62^#(mark(X1), X2)	→	U62^#(X1, X2)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
U61^#(X1, mark(X2), X3)	→	U61^#(X1, X2, X3)	isNatList^#(active(X))	→	isNatList^#(X)
U41^#(X1, mark(X2))	→	U41^#(X1, X2)	isNatIList^#(active(X))	→	isNatIList^#(X)
active^#(isNat(0))	→	mark^#(tt)	active^#(U61(tt, L, N))	→	isNat^#(N)
isNatIList^#(mark(X))	→	isNatIList^#(X)	mark^#(isNatIList(X))	→	isNatIList^#(X)
mark^#(U41(X1, X2))	→	U41^#(mark(X1), X2)	cons^#(X1, active(X2))	→	cons^#(X1, X2)
active^#(U62(tt, L))	→	mark^#(s(length(L)))	active^#(U52(tt))	→	mark^#(tt)
active^#(isNatList(cons(V1, V2)))	→	isNat^#(V1)	U31^#(mark(X))	→	U31^#(X)
mark^#(U62(X1, X2))	→	U62^#(mark(X1), X2)	mark^#(isNat(X))	→	active^#(isNat(X))
mark^#(U11(X))	→	U11^#(mark(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(U61(X1, X2, X3))	→	mark^#(X1)	mark^#(U42(X))	→	mark^#(X)
U51^#(active(X1), X2)	→	U51^#(X1, X2)	mark^#(isNat(X))	→	isNat^#(X)
isNat^#(mark(X))	→	isNat^#(X)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	active^#(length(cons(N, L)))	→	isNatList^#(L)
active^#(isNatList(nil))	→	mark^#(tt)	mark^#(nil)	→	active^#(nil)
active^#(U62(tt, L))	→	s^#(length(L))	active^#(isNatIList(zeros))	→	mark^#(tt)
U41^#(mark(X1), X2)	→	U41^#(X1, X2)	mark^#(cons(X1, X2))	→	cons^#(mark(X1), X2)
mark^#(length(X))	→	length^#(mark(X))	U31^#(active(X))	→	U31^#(X)
U21^#(mark(X))	→	U21^#(X)	U21^#(active(X))	→	U21^#(X)
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(isNatIList(V))	→	isNatList^#(V)	U61^#(X1, X2, active(X3))	→	U61^#(X1, X2, X3)
s^#(mark(X))	→	s^#(X)	U52^#(mark(X))	→	U52^#(X)
active^#(zeros)	→	mark^#(cons(0, zeros))	s^#(active(X))	→	s^#(X)
active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))	mark^#(length(X))	→	active^#(length(mark(X)))
mark^#(isNatList(X))	→	active^#(isNatList(X))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	mark^#(U52(X)) → U52^#(mark(X))
active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))	mark^#(s(X)) → mark^#(X)
mark^#(isNatIList(X)) → active^#(isNatIList(X))	active^#(isNat(0)) → mark^#(tt)
mark^#(cons(X1, X2)) → mark^#(X1)	mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))
mark^#(U52(X)) → mark^#(X)	active^#(U31(tt)) → mark^#(tt)
mark^#(U42(X)) → active^#(U42(mark(X)))	active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))
active^#(U62(tt, L)) → mark^#(s(length(L)))	mark^#(U31(X)) → mark^#(X)
active^#(U52(tt)) → mark^#(tt)	active^#(U42(tt)) → mark^#(tt)
mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))	mark^#(isNat(X)) → active^#(isNat(X))
mark^#(tt) → active^#(tt)	mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))
mark^#(U61(X1, X2, X3)) → mark^#(X1)	mark^#(U11(X)) → mark^#(X)
mark^#(U42(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → mark^#(X1)	mark^#(U21(X)) → mark^#(X)
active^#(U11(tt)) → mark^#(tt)	mark^#(U52(X)) → active^#(U52(mark(X)))
active^#(isNatList(nil)) → mark^#(tt)	mark^#(nil) → active^#(nil)
mark^#(length(X)) → mark^#(X)	active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))
mark^#(U41(X1, X2)) → mark^#(X1)	mark^#(U11(X)) → active^#(U11(mark(X)))
active^#(isNatIList(zeros)) → mark^#(tt)	mark^#(zeros) → active^#(zeros)
mark^#(0) → active^#(0)	active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))
mark^#(s(X)) → active^#(s(mark(X)))	mark^#(U31(X)) → active^#(U31(mark(X)))
active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))	active^#(zeros) → mark^#(cons(0, zeros))
active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))	mark^#(isNatList(X)) → active^#(isNatList(X))
mark^#(length(X)) → active^#(length(mark(X)))	active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))
mark^#(tt) → active^#(tt)	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
active^#(isNat(length(V1))) → U11^#(isNatList(V1))	mark^#(nil) → active^#(nil)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	mark^#(length(X)) → mark^#(X)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
active^#(isNatIList(zeros)) → mark^#(tt)	mark^#(zeros) → active^#(zeros)
mark^#(0) → active^#(0)	active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))
mark^#(s(X)) → active^#(s(mark(X)))	mark^#(U31(X)) → active^#(U31(mark(X)))
active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))	active^#(zeros) → mark^#(cons(0, zeros))
active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))	mark^#(isNatList(X)) → active^#(isNatList(X))
mark^#(length(X)) → active^#(length(mark(X)))	active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

isNatIList^#(active(X)) → isNatIList^#(X)

isNatIList^#(mark(X)) → isNatIList^#(X)

length^#(mark(X)) → length^#(X)

length^#(active(X)) → length^#(X)

U41^#(X1, mark(X2)) → U41^#(X1, X2)	U41^#(X1, active(X2)) → U41^#(X1, X2)
U41^#(mark(X1), X2) → U41^#(X1, X2)	U41^#(active(X1), X2) → U41^#(X1, X2)

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U42(X)) → mark^#(X)
mark^#(U62(X1, X2)) → mark^#(X1)	mark^#(U51(X1, X2)) → mark^#(X1)
active^#(U11(tt)) → mark^#(tt)	mark^#(U21(X)) → mark^#(X)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))
mark^#(length(X)) → mark^#(X)	mark^#(U11(X)) → active^#(U11(mark(X)))
mark^#(U41(X1, X2)) → mark^#(X1)	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
mark^#(s(X)) → active^#(s(mark(X)))	active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	active^#(isNatIList(V)) → U31^#(isNatList(V))
mark^#(length(X)) → active^#(length(mark(X)))	active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

U62^#(X1, active(X2)) → U62^#(X1, X2)	U62^#(X1, mark(X2)) → U62^#(X1, X2)
U62^#(active(X1), X2) → U62^#(X1, X2)	U62^#(mark(X1), X2) → U62^#(X1, X2)

cons^#(X1, active(X2)) → cons^#(X1, X2)	cons^#(mark(X1), X2) → cons^#(X1, X2)
cons^#(active(X1), X2) → cons^#(X1, X2)	cons^#(X1, mark(X2)) → cons^#(X1, X2)

U61^#(X1, X2, active(X3)) → U61^#(X1, X2, X3)	U61^#(X1, X2, mark(X3)) → U61^#(X1, X2, X3)
U61^#(active(X1), X2, X3) → U61^#(X1, X2, X3)	U61^#(mark(X1), X2, X3) → U61^#(X1, X2, X3)
U61^#(X1, mark(X2), X3) → U61^#(X1, X2, X3)	U61^#(X1, active(X2), X3) → U61^#(X1, X2, X3)

isNat^#(active(X)) → isNat^#(X)

isNat^#(mark(X)) → isNat^#(X)

U42^#(mark(X)) → U42^#(X)

U42^#(active(X)) → U42^#(X)

U52^#(active(X)) → U52^#(X)

U52^#(mark(X)) → U52^#(X)

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))
mark^#(tt) → active^#(tt)	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U42(X)) → mark^#(X)
mark^#(U62(X1, X2)) → mark^#(X1)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
active^#(U41(tt, V2)) → U42^#(isNatIList(V2))	mark^#(nil) → active^#(nil)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	mark^#(length(X)) → mark^#(X)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
active^#(isNatIList(zeros)) → mark^#(tt)	mark^#(zeros) → active^#(zeros)
mark^#(0) → active^#(0)	active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))
mark^#(s(X)) → active^#(s(mark(X)))	mark^#(U31(X)) → active^#(U31(mark(X)))
active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))	active^#(zeros) → mark^#(cons(0, zeros))
active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))	mark^#(isNatList(X)) → active^#(isNatList(X))
mark^#(length(X)) → active^#(length(mark(X)))	active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

U11^#(active(X)) → U11^#(X)

U11^#(mark(X)) → U11^#(X)

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U42(X)) → mark^#(X)
mark^#(U62(X1, X2)) → mark^#(X1)	mark^#(U51(X1, X2)) → mark^#(X1)
active^#(U11(tt)) → mark^#(tt)	mark^#(U21(X)) → mark^#(X)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	mark^#(U11(X)) → active^#(U11(mark(X)))
mark^#(U41(X1, X2)) → mark^#(X1)	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(length(X)) → active^#(length(mark(X)))	mark^#(isNatList(X)) → active^#(isNatList(X))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

isNatList^#(mark(X)) → isNatList^#(X)

isNatList^#(active(X)) → isNatList^#(X)

U31^#(active(X)) → U31^#(X)

U31^#(mark(X)) → U31^#(X)

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → U62^#(mark(X1), X2)
mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))	mark^#(isNat(X)) → active^#(isNat(X))
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → mark^#(X1)	mark^#(U11(X)) → mark^#(X)
mark^#(U62(X1, X2)) → mark^#(X1)	mark^#(U42(X)) → mark^#(X)
mark^#(U51(X1, X2)) → mark^#(X1)	mark^#(U21(X)) → mark^#(X)
active^#(U11(tt)) → mark^#(tt)	mark^#(U52(X)) → active^#(U52(mark(X)))
active^#(isNatList(nil)) → mark^#(tt)	mark^#(nil) → active^#(nil)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	mark^#(length(X)) → mark^#(X)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
active^#(isNatIList(zeros)) → mark^#(tt)	mark^#(zeros) → active^#(zeros)
mark^#(0) → active^#(0)	active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))
mark^#(s(X)) → active^#(s(mark(X)))	mark^#(U31(X)) → active^#(U31(mark(X)))
active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))	active^#(zeros) → mark^#(cons(0, zeros))
active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))	mark^#(length(X)) → active^#(length(mark(X)))
mark^#(isNatList(X)) → active^#(isNatList(X))	active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

s^#(mark(X)) → s^#(X)

s^#(active(X)) → s^#(X)

U51^#(X1, mark(X2)) → U51^#(X1, X2)	U51^#(mark(X1), X2) → U51^#(X1, X2)
U51^#(X1, active(X2)) → U51^#(X1, X2)	U51^#(active(X1), X2) → U51^#(X1, X2)

U21^#(mark(X)) → U21^#(X)

U21^#(active(X)) → U21^#(X)

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(isNatList(cons(V1, V2))) → isNat^#(V1)	active^#(U42(tt)) → mark^#(tt)
mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))	mark^#(isNat(X)) → active^#(isNat(X))
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → mark^#(X1)	mark^#(U11(X)) → mark^#(X)
mark^#(U62(X1, X2)) → mark^#(X1)	mark^#(U42(X)) → mark^#(X)
mark^#(U51(X1, X2)) → mark^#(X1)	mark^#(U21(X)) → mark^#(X)
active^#(U11(tt)) → mark^#(tt)	mark^#(U52(X)) → active^#(U52(mark(X)))
active^#(isNatList(nil)) → mark^#(tt)	mark^#(nil) → active^#(nil)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	mark^#(length(X)) → mark^#(X)
active^#(isNatIList(zeros)) → mark^#(tt)	mark^#(U11(X)) → active^#(U11(mark(X)))
mark^#(U41(X1, X2)) → mark^#(X1)	mark^#(zeros) → active^#(zeros)
mark^#(0) → active^#(0)	active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))
mark^#(s(X)) → active^#(s(mark(X)))	mark^#(U31(X)) → active^#(U31(mark(X)))
active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))	active^#(zeros) → mark^#(cons(0, zeros))
active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))	mark^#(length(X)) → active^#(length(mark(X)))
mark^#(isNatList(X)) → active^#(isNatList(X))	active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U61^#(X1, X2, active(X3))	→	U61^#(X1, X2, X3)	U61^#(X1, X2, mark(X3))	→	U61^#(X1, X2, X3)
U61^#(active(X1), X2, X3)	→	U61^#(X1, X2, X3)	U61^#(mark(X1), X2, X3)	→	U61^#(X1, X2, X3)
U61^#(X1, mark(X2), X3)	→	U61^#(X1, X2, X3)	U61^#(X1, active(X2), X3)	→	U61^#(X1, X2, X3)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U61#): 1

Thus, the following dependency pairs are removed:

U61^#(active(X1), X2, X3)

→

U61^#(X1, X2, X3)

U61^#(mark(X1), X2, X3)

→

U61^#(X1, X2, X3)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

cons^#(X1, active(X2))	→	cons^#(X1, X2)		cons^#(mark(X1), X2)	→	cons^#(X1, X2)
cons^#(active(X1), X2)	→	cons^#(X1, X2)		cons^#(X1, mark(X2))	→	cons^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (cons#): 1

Thus, the following dependency pairs are removed:

cons^#(mark(X1), X2)

→

cons^#(X1, X2)

cons^#(active(X1), X2)

→

cons^#(X1, X2)

Problem 4: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(isNatList(cons(V1, V2)))	→	isNat^#(V1)	active^#(U42(tt))	→	mark^#(tt)
mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))	mark^#(isNat(X))	→	active^#(isNat(X))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(U61(X1, X2, X3))	→	mark^#(X1)	mark^#(U11(X))	→	mark^#(X)
mark^#(U42(X))	→	mark^#(X)	mark^#(U62(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	mark^#(X1)	active^#(U11(tt))	→	mark^#(tt)
mark^#(U21(X))	→	mark^#(X)	mark^#(U52(X))	→	active^#(U52(mark(X)))
active^#(isNatList(nil))	→	mark^#(tt)	mark^#(nil)	→	active^#(nil)
mark^#(length(X))	→	mark^#(X)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
mark^#(U41(X1, X2))	→	mark^#(X1)	mark^#(U11(X))	→	active^#(U11(mark(X)))
active^#(isNatIList(zeros))	→	mark^#(tt)	mark^#(zeros)	→	active^#(zeros)
mark^#(0)	→	active^#(0)	mark^#(s(X))	→	active^#(s(mark(X)))
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(U31(X))	→	active^#(U31(mark(X)))
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	active^#(zeros)	→	mark^#(cons(0, zeros))
active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))	mark^#(length(X))	→	active^#(length(mark(X)))
mark^#(isNatList(X))	→	active^#(isNatList(X))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	mark^#(length(X)) → active^#(length(mark(X)))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

isNatIList^#(active(X))

→

isNatIList^#(X)

isNatIList^#(mark(X))

→

isNatIList^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (isNatIList#): 1

Thus, the following dependency pairs are removed:

isNatIList^#(active(X))

→

isNatIList^#(X)

isNatIList^#(mark(X))

→

isNatIList^#(X)

Problem 6: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	U62^#(mark(X1), X2)
mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))	mark^#(isNat(X))	→	active^#(isNat(X))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(U61(X1, X2, X3))	→	mark^#(X1)	mark^#(U11(X))	→	mark^#(X)
mark^#(U42(X))	→	mark^#(X)	mark^#(U62(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	mark^#(X1)	active^#(U11(tt))	→	mark^#(tt)
mark^#(U21(X))	→	mark^#(X)	mark^#(U52(X))	→	active^#(U52(mark(X)))
active^#(isNatList(nil))	→	mark^#(tt)	mark^#(nil)	→	active^#(nil)
mark^#(length(X))	→	mark^#(X)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
active^#(isNatIList(zeros))	→	mark^#(tt)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	mark^#(zeros)	→	active^#(zeros)
mark^#(0)	→	active^#(0)	mark^#(s(X))	→	active^#(s(mark(X)))
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(U31(X))	→	active^#(U31(mark(X)))
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	active^#(zeros)	→	mark^#(cons(0, zeros))
active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))	mark^#(length(X))	→	active^#(length(mark(X)))
mark^#(isNatList(X))	→	active^#(isNatList(X))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	mark^#(length(X)) → active^#(length(mark(X)))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

isNatList^#(mark(X))

→

isNatList^#(X)

isNatList^#(active(X))

→

isNatList^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (isNatList#): 1

Thus, the following dependency pairs are removed:

isNatList^#(mark(X))

→

isNatList^#(X)

isNatList^#(active(X))

→

isNatList^#(X)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U11^#(active(X))

→

U11^#(X)

U11^#(mark(X))

→

U11^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U11#): 1

Thus, the following dependency pairs are removed:

U11^#(active(X))

→

U11^#(X)

U11^#(mark(X))

→

U11^#(X)

Problem 9: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(tt)	→	active^#(tt)
mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U62(X1, X2))	→	mark^#(X1)
mark^#(U42(X))	→	mark^#(X)	mark^#(U51(X1, X2))	→	mark^#(X1)
mark^#(U21(X))	→	mark^#(X)	active^#(U11(tt))	→	mark^#(tt)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	mark^#(length(X))	→	mark^#(X)
active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros))	→	mark^#(tt)
mark^#(U41(X1, X2))	→	mark^#(X1)	mark^#(U11(X))	→	active^#(U11(mark(X)))
mark^#(zeros)	→	active^#(zeros)	mark^#(0)	→	active^#(0)
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(s(X))	→	active^#(s(mark(X)))
mark^#(U31(X))	→	active^#(U31(mark(X)))	active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))
active^#(zeros)	→	mark^#(cons(0, zeros))	active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))
mark^#(isNatList(X))	→	active^#(isNatList(X))	active^#(isNatIList(V))	→	U31^#(isNatList(V))
mark^#(length(X))	→	active^#(length(mark(X)))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	mark^#(length(X)) → active^#(length(mark(X)))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 10: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U21^#(mark(X))

→

U21^#(X)

U21^#(active(X))

→

U21^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U21#): 1

Thus, the following dependency pairs are removed:

U21^#(mark(X))

→

U21^#(X)

U21^#(active(X))

→

U21^#(X)

Problem 11: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U42(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
active^#(isNat(length(V1)))	→	U11^#(isNatList(V1))	mark^#(nil)	→	active^#(nil)
mark^#(length(X))	→	mark^#(X)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
active^#(isNatIList(zeros))	→	mark^#(tt)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	mark^#(zeros)	→	active^#(zeros)
mark^#(0)	→	active^#(0)	mark^#(s(X))	→	active^#(s(mark(X)))
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(U31(X))	→	active^#(U31(mark(X)))
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	active^#(zeros)	→	mark^#(cons(0, zeros))
active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
mark^#(length(X))	→	active^#(length(mark(X)))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	mark^#(length(X)) → active^#(length(mark(X)))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 12: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s^#(mark(X))

→

s^#(X)

s^#(active(X))

→

s^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (s#): 1

Thus, the following dependency pairs are removed:

s^#(mark(X))

→

s^#(X)

s^#(active(X))

→

s^#(X)

Problem 14: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))
mark^#(s(X))	→	mark^#(X)	mark^#(isNatIList(X))	→	active^#(isNatIList(X))
active^#(isNat(0))	→	mark^#(tt)	mark^#(cons(X1, X2))	→	mark^#(X1)
mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))	mark^#(U52(X))	→	mark^#(X)
active^#(U31(tt))	→	mark^#(tt)	mark^#(U42(X))	→	active^#(U42(mark(X)))
active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L))	→	mark^#(s(length(L)))
mark^#(U31(X))	→	mark^#(X)	active^#(U52(tt))	→	mark^#(tt)
active^#(U42(tt))	→	mark^#(tt)	mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))
mark^#(isNat(X))	→	active^#(isNat(X))	mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))
mark^#(tt)	→	active^#(tt)	mark^#(U61(X1, X2, X3))	→	mark^#(X1)
mark^#(U11(X))	→	mark^#(X)	mark^#(U62(X1, X2))	→	mark^#(X1)
mark^#(U42(X))	→	mark^#(X)	mark^#(U51(X1, X2))	→	mark^#(X1)
active^#(U11(tt))	→	mark^#(tt)	mark^#(U21(X))	→	mark^#(X)
mark^#(U52(X))	→	active^#(U52(mark(X)))	active^#(isNatList(nil))	→	mark^#(tt)
mark^#(nil)	→	active^#(nil)	active^#(U41(tt, V2))	→	U42^#(isNatIList(V2))
mark^#(length(X))	→	mark^#(X)	active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))
active^#(isNatIList(zeros))	→	mark^#(tt)	mark^#(U41(X1, X2))	→	mark^#(X1)
mark^#(U11(X))	→	active^#(U11(mark(X)))	mark^#(zeros)	→	active^#(zeros)
mark^#(0)	→	active^#(0)	mark^#(s(X))	→	active^#(s(mark(X)))
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(U31(X))	→	active^#(U31(mark(X)))
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	active^#(zeros)	→	mark^#(cons(0, zeros))
active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
mark^#(length(X))	→	active^#(length(mark(X)))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	mark^#(length(X)) → active^#(length(mark(X)))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 15: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U42^#(mark(X))

→

U42^#(X)

U42^#(active(X))

→

U42^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U42#): 1

Thus, the following dependency pairs are removed:

U42^#(mark(X))

→

U42^#(X)

U42^#(active(X))

→

U42^#(X)

Problem 16: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U31^#(active(X))

→

U31^#(X)

U31^#(mark(X))

→

U31^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U31#): 1

Thus, the following dependency pairs are removed:

U31^#(active(X))

→

U31^#(X)

U31^#(mark(X))

→

U31^#(X)

Problem 17: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark^#(cons(X1, X2))	→	active^#(cons(mark(X1), X2))	mark^#(U21(X))	→	active^#(U21(mark(X)))
active^#(U21(tt))	→	mark^#(tt)	active^#(U41(tt, V2))	→	mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	active^#(length(nil))	→	mark^#(0)
mark^#(U41(X1, X2))	→	active^#(U41(mark(X1), X2))	mark^#(U52(X))	→	U52^#(mark(X))
active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))	mark^#(s(X))	→	mark^#(X)
mark^#(isNatIList(X))	→	active^#(isNatIList(X))	active^#(isNat(0))	→	mark^#(tt)
mark^#(cons(X1, X2))	→	mark^#(X1)	mark^#(U51(X1, X2))	→	active^#(U51(mark(X1), X2))
mark^#(U52(X))	→	mark^#(X)	active^#(U31(tt))	→	mark^#(tt)
mark^#(U42(X))	→	active^#(U42(mark(X)))	active^#(length(cons(N, L)))	→	mark^#(U61(isNatList(L), L, N))
active^#(U62(tt, L))	→	mark^#(s(length(L)))	mark^#(U31(X))	→	mark^#(X)
active^#(U52(tt))	→	mark^#(tt)	active^#(U42(tt))	→	mark^#(tt)
mark^#(U62(X1, X2))	→	active^#(U62(mark(X1), X2))	mark^#(isNat(X))	→	active^#(isNat(X))
mark^#(U61(X1, X2, X3))	→	active^#(U61(mark(X1), X2, X3))	mark^#(tt)	→	active^#(tt)
mark^#(U61(X1, X2, X3))	→	mark^#(X1)	mark^#(U11(X))	→	mark^#(X)
mark^#(U62(X1, X2))	→	mark^#(X1)	mark^#(U42(X))	→	mark^#(X)
mark^#(U51(X1, X2))	→	mark^#(X1)	active^#(U11(tt))	→	mark^#(tt)
mark^#(U21(X))	→	mark^#(X)	mark^#(U52(X))	→	active^#(U52(mark(X)))
active^#(isNatList(nil))	→	mark^#(tt)	mark^#(nil)	→	active^#(nil)
active^#(isNatList(cons(V1, V2)))	→	mark^#(U51(isNat(V1), V2))	mark^#(length(X))	→	mark^#(X)
active^#(isNatIList(zeros))	→	mark^#(tt)	mark^#(U11(X))	→	active^#(U11(mark(X)))
mark^#(U41(X1, X2))	→	mark^#(X1)	mark^#(zeros)	→	active^#(zeros)
mark^#(0)	→	active^#(0)	mark^#(s(X))	→	active^#(s(mark(X)))
active^#(isNatIList(cons(V1, V2)))	→	mark^#(U41(isNat(V1), V2))	mark^#(U31(X))	→	active^#(U31(mark(X)))
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	active^#(zeros)	→	mark^#(cons(0, zeros))
active^#(U51(tt, V2))	→	mark^#(U52(isNatList(V2)))	mark^#(isNatList(X))	→	active^#(isNatList(X))
mark^#(length(X))	→	active^#(length(mark(X)))	active^#(U61(tt, L, N))	→	mark^#(U62(isNat(N), L))

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

The following SCCs where found

mark^#(cons(X1, X2)) → active^#(cons(mark(X1), X2))	mark^#(U21(X)) → active^#(U21(mark(X)))
active^#(U21(tt)) → mark^#(tt)	active^#(U41(tt, V2)) → mark^#(U42(isNatIList(V2)))
active^#(isNat(length(V1))) → mark^#(U11(isNatList(V1)))	active^#(length(nil)) → mark^#(0)
mark^#(U41(X1, X2)) → active^#(U41(mark(X1), X2))	active^#(isNatIList(V)) → mark^#(U31(isNatList(V)))
mark^#(s(X)) → mark^#(X)	mark^#(isNatIList(X)) → active^#(isNatIList(X))
active^#(isNat(0)) → mark^#(tt)	mark^#(cons(X1, X2)) → mark^#(X1)
mark^#(U51(X1, X2)) → active^#(U51(mark(X1), X2))	mark^#(U52(X)) → mark^#(X)
active^#(U31(tt)) → mark^#(tt)	mark^#(U42(X)) → active^#(U42(mark(X)))
active^#(length(cons(N, L))) → mark^#(U61(isNatList(L), L, N))	active^#(U62(tt, L)) → mark^#(s(length(L)))
mark^#(U31(X)) → mark^#(X)	active^#(U52(tt)) → mark^#(tt)
active^#(U42(tt)) → mark^#(tt)	mark^#(U62(X1, X2)) → active^#(U62(mark(X1), X2))
mark^#(isNat(X)) → active^#(isNat(X))	mark^#(tt) → active^#(tt)
mark^#(U61(X1, X2, X3)) → active^#(U61(mark(X1), X2, X3))	mark^#(U61(X1, X2, X3)) → mark^#(X1)
mark^#(U11(X)) → mark^#(X)	mark^#(U62(X1, X2)) → mark^#(X1)
mark^#(U42(X)) → mark^#(X)	mark^#(U51(X1, X2)) → mark^#(X1)
mark^#(U21(X)) → mark^#(X)	active^#(U11(tt)) → mark^#(tt)
mark^#(U52(X)) → active^#(U52(mark(X)))	active^#(isNatList(nil)) → mark^#(tt)
mark^#(nil) → active^#(nil)	mark^#(length(X)) → mark^#(X)
active^#(isNatList(cons(V1, V2))) → mark^#(U51(isNat(V1), V2))	active^#(isNatIList(zeros)) → mark^#(tt)
mark^#(U11(X)) → active^#(U11(mark(X)))	mark^#(U41(X1, X2)) → mark^#(X1)
mark^#(zeros) → active^#(zeros)	mark^#(0) → active^#(0)
active^#(isNatIList(cons(V1, V2))) → mark^#(U41(isNat(V1), V2))	mark^#(s(X)) → active^#(s(mark(X)))
mark^#(U31(X)) → active^#(U31(mark(X)))	active^#(isNat(s(V1))) → mark^#(U21(isNat(V1)))
active^#(zeros) → mark^#(cons(0, zeros))	active^#(U51(tt, V2)) → mark^#(U52(isNatList(V2)))
mark^#(isNatList(X)) → active^#(isNatList(X))	mark^#(length(X)) → active^#(length(mark(X)))
active^#(U61(tt, L, N)) → mark^#(U62(isNat(N), L))

Problem 18: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U51^#(X1, mark(X2))	→	U51^#(X1, X2)		U51^#(mark(X1), X2)	→	U51^#(X1, X2)
U51^#(X1, active(X2))	→	U51^#(X1, X2)		U51^#(active(X1), X2)	→	U51^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U51#): 1

Thus, the following dependency pairs are removed:

U51^#(mark(X1), X2)

→

U51^#(X1, X2)

U51^#(active(X1), X2)

→

U51^#(X1, X2)

Problem 19: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

isNat^#(active(X))

→

isNat^#(X)

isNat^#(mark(X))

→

isNat^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (isNat#): 1

Thus, the following dependency pairs are removed:

isNat^#(active(X))

→

isNat^#(X)

isNat^#(mark(X))

→

isNat^#(X)

Problem 20: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

length^#(mark(X))

→

length^#(X)

length^#(active(X))

→

length^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (length#): 1

Thus, the following dependency pairs are removed:

length^#(mark(X))

→

length^#(X)

length^#(active(X))

→

length^#(X)

Problem 21: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U62^#(X1, active(X2))	→	U62^#(X1, X2)		U62^#(X1, mark(X2))	→	U62^#(X1, X2)
U62^#(active(X1), X2)	→	U62^#(X1, X2)		U62^#(mark(X1), X2)	→	U62^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U62#): 1

Thus, the following dependency pairs are removed:

U62^#(mark(X1), X2)

→

U62^#(X1, X2)

U62^#(active(X1), X2)

→

U62^#(X1, X2)

Problem 22: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U52^#(active(X))

→

U52^#(X)

U52^#(mark(X))

→

U52^#(X)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U52#): 1

Thus, the following dependency pairs are removed:

U52^#(active(X))

→

U52^#(X)

U52^#(mark(X))

→

U52^#(X)

Problem 23: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

U41^#(X1, mark(X2))	→	U41^#(X1, X2)		U41^#(X1, active(X2))	→	U41^#(X1, X2)
U41^#(mark(X1), X2)	→	U41^#(X1, X2)		U41^#(active(X1), X2)	→	U41^#(X1, X2)

Rewrite Rules

active(zeros)	→	mark(cons(0, zeros))	active(U11(tt))	→	mark(tt)
active(U21(tt))	→	mark(tt)	active(U31(tt))	→	mark(tt)
active(U41(tt, V2))	→	mark(U42(isNatIList(V2)))	active(U42(tt))	→	mark(tt)
active(U51(tt, V2))	→	mark(U52(isNatList(V2)))	active(U52(tt))	→	mark(tt)
active(U61(tt, L, N))	→	mark(U62(isNat(N), L))	active(U62(tt, L))	→	mark(s(length(L)))
active(isNat(0))	→	mark(tt)	active(isNat(length(V1)))	→	mark(U11(isNatList(V1)))
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	active(isNatIList(V))	→	mark(U31(isNatList(V)))
active(isNatIList(zeros))	→	mark(tt)	active(isNatIList(cons(V1, V2)))	→	mark(U41(isNat(V1), V2))
active(isNatList(nil))	→	mark(tt)	active(isNatList(cons(V1, V2)))	→	mark(U51(isNat(V1), V2))
active(length(nil))	→	mark(0)	active(length(cons(N, L)))	→	mark(U61(isNatList(L), L, N))
mark(zeros)	→	active(zeros)	mark(cons(X1, X2))	→	active(cons(mark(X1), X2))
mark(0)	→	active(0)	mark(U11(X))	→	active(U11(mark(X)))
mark(tt)	→	active(tt)	mark(U21(X))	→	active(U21(mark(X)))
mark(U31(X))	→	active(U31(mark(X)))	mark(U41(X1, X2))	→	active(U41(mark(X1), X2))
mark(U42(X))	→	active(U42(mark(X)))	mark(isNatIList(X))	→	active(isNatIList(X))
mark(U51(X1, X2))	→	active(U51(mark(X1), X2))	mark(U52(X))	→	active(U52(mark(X)))
mark(isNatList(X))	→	active(isNatList(X))	mark(U61(X1, X2, X3))	→	active(U61(mark(X1), X2, X3))
mark(U62(X1, X2))	→	active(U62(mark(X1), X2))	mark(isNat(X))	→	active(isNat(X))
mark(s(X))	→	active(s(mark(X)))	mark(length(X))	→	active(length(mark(X)))
mark(nil)	→	active(nil)	cons(mark(X1), X2)	→	cons(X1, X2)
cons(X1, mark(X2))	→	cons(X1, X2)	cons(active(X1), X2)	→	cons(X1, X2)
cons(X1, active(X2))	→	cons(X1, X2)	U11(mark(X))	→	U11(X)
U11(active(X))	→	U11(X)	U21(mark(X))	→	U21(X)
U21(active(X))	→	U21(X)	U31(mark(X))	→	U31(X)
U31(active(X))	→	U31(X)	U41(mark(X1), X2)	→	U41(X1, X2)
U41(X1, mark(X2))	→	U41(X1, X2)	U41(active(X1), X2)	→	U41(X1, X2)
U41(X1, active(X2))	→	U41(X1, X2)	U42(mark(X))	→	U42(X)
U42(active(X))	→	U42(X)	isNatIList(mark(X))	→	isNatIList(X)
isNatIList(active(X))	→	isNatIList(X)	U51(mark(X1), X2)	→	U51(X1, X2)
U51(X1, mark(X2))	→	U51(X1, X2)	U51(active(X1), X2)	→	U51(X1, X2)
U51(X1, active(X2))	→	U51(X1, X2)	U52(mark(X))	→	U52(X)
U52(active(X))	→	U52(X)	isNatList(mark(X))	→	isNatList(X)
isNatList(active(X))	→	isNatList(X)	U61(mark(X1), X2, X3)	→	U61(X1, X2, X3)
U61(X1, mark(X2), X3)	→	U61(X1, X2, X3)	U61(X1, X2, mark(X3))	→	U61(X1, X2, X3)
U61(active(X1), X2, X3)	→	U61(X1, X2, X3)	U61(X1, active(X2), X3)	→	U61(X1, X2, X3)
U61(X1, X2, active(X3))	→	U61(X1, X2, X3)	U62(mark(X1), X2)	→	U62(X1, X2)
U62(X1, mark(X2))	→	U62(X1, X2)	U62(active(X1), X2)	→	U62(X1, X2)
U62(X1, active(X2))	→	U62(X1, X2)	isNat(mark(X))	→	isNat(X)
isNat(active(X))	→	isNat(X)	s(mark(X))	→	s(X)
s(active(X))	→	s(X)	length(mark(X))	→	length(X)
length(active(X))	→	length(X)

Original Signature

Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, U31, U21, cons, nil

Strategy

Projection

The following projection was used:

π (U41#): 1

Thus, the following dependency pairs are removed:

U41^#(mark(X1), X2)

→

U41^#(X1, X2)

U41^#(active(X1), X2)

→

U41^#(X1, X2)