Termination Proof using AProVE

Term Rewriting System R:
[X, L, X1, X2]
active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(incr(cons(X, L))) -> CONS(s(X), incr(L))
ACTIVE(incr(cons(X, L))) -> S(X)
ACTIVE(incr(cons(X, L))) -> INCR(L)
ACTIVE(adx(cons(X, L))) -> INCR(cons(X, adx(L)))
ACTIVE(adx(cons(X, L))) -> CONS(X, adx(L))
ACTIVE(adx(cons(X, L))) -> ADX(L)
ACTIVE(nats) -> ADX(zeros)
ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(incr(X)) -> INCR(active(X))
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ADX(active(X))
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> HEAD(active(X))
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(tail(X)) -> TAIL(active(X))
ACTIVE(tail(X)) -> ACTIVE(X)
INCR(mark(X)) -> INCR(X)
INCR(ok(X)) -> INCR(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
ADX(mark(X)) -> ADX(X)
ADX(ok(X)) -> ADX(X)
HEAD(mark(X)) -> HEAD(X)
HEAD(ok(X)) -> HEAD(X)
TAIL(mark(X)) -> TAIL(X)
TAIL(ok(X)) -> TAIL(X)
PROPER(incr(X)) -> INCR(proper(X))
PROPER(incr(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(adx(X)) -> ADX(proper(X))
PROPER(adx(X)) -> PROPER(X)
PROPER(head(X)) -> HEAD(proper(X))
PROPER(head(X)) -> PROPER(X)
PROPER(tail(X)) -> TAIL(proper(X))
PROPER(tail(X)) -> PROPER(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains nine SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(tail(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(adx(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R could not be shown.
Duration:
0:00 minutes