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))

Innermost Termination of R to be shown. 



   R
     ↳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. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  0  
  POL(ok(x1))=  1 + x1  
  POL(CONS(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 10
             ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

CONS(mark(X1), X2) -> CONS(X1, X2)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(CONS(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 10
             ↳Polo
             ...
               →DP Problem 11
                 ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

S(mark(X)) -> S(X)
S(ok(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

S(mark(X)) -> S(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(ok(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 12
             ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

S(ok(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

S(ok(X)) -> S(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 12
             ↳Polo
             ...
               →DP Problem 13
                 ↳Dependency Graph
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

INCR(ok(X)) -> INCR(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(INCR(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 14
             ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

INCR(mark(X)) -> INCR(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(INCR(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 14
             ↳Polo
             ...
               →DP Problem 15
                 ↳Dependency Graph
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ADX(ok(X)) -> ADX(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ADX(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 16
             ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ADX(mark(X)) -> ADX(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ADX(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 16
             ↳Polo
             ...
               →DP Problem 17
                 ↳Dependency Graph
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

HEAD(ok(X)) -> HEAD(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(HEAD(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 18
             ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

HEAD(mark(X)) -> HEAD(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(HEAD(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 18
             ↳Polo
             ...
               →DP Problem 19
                 ↳Dependency Graph
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

TAIL(ok(X)) -> TAIL(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TAIL(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 20
             ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

TAIL(mark(X)) -> TAIL(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TAIL(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 20
             ↳Polo
             ...
               →DP Problem 21
                 ↳Dependency Graph
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(tail(X)) -> ACTIVE(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(tail(x1))=  1 + x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  
  POL(head(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 22
             ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(head(X)) -> ACTIVE(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  
  POL(head(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 22
             ↳Polo
             ...
               →DP Problem 23
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(adx(X)) -> ACTIVE(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(adx(x1))=  1 + x1  
  POL(cons(x1, x2))=  x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 22
             ↳Polo
             ...
               →DP Problem 24
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(s(X)) -> ACTIVE(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 22
             ↳Polo
             ...
               →DP Problem 25
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(cons(X1, X2)) -> ACTIVE(X1)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1  
  POL(incr(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 22
             ↳Polo
             ...
               →DP Problem 26
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(incr(X)) -> ACTIVE(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(incr(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 22
             ↳Polo
             ...
               →DP Problem 27
                 ↳Dependency Graph
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(tail(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  1 + x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  
  POL(head(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(head(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  
  POL(head(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 29
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(adx(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(adx(x1))=  1 + x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 30
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(s(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(incr(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 31
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(incr(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 32
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(incr(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(incr(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 33
                 ↳Dependency Graph
       →DP Problem 9
         ↳Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining Obligation(s)




The following remains to be proven:
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))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes