Term Rewriting System R:
[X, Y, Z, X1, X2, X3]
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(first(s(X), cons(Y, Z))) -> CONS(Y, first(X, Z))
ACTIVE(first(s(X), cons(Y, Z))) -> FIRST(X, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(and(X1, X2)) -> AND(active(X1), X2)
ACTIVE(and(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(active(X1), X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(X1, active(X2))
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
AND(mark(X1), X2) -> AND(X1, X2)
AND(ok(X1), ok(X2)) -> AND(X1, X2)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
PROPER(and(X1, X2)) -> AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> IF(proper(X1), proper(X2), proper(X3))
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
S(ok(X)) -> S(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(ok(X)) -> FROM(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 10 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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 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 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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(ADD(x1, x2))=  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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 14
             ↳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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST(x1, x2))=  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 15
             ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST(x1, x2))=  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 15
             ↳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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST(x1, x2))=  x2  
  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 15
             ↳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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

FROM(ok(X)) -> FROM(X)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

FROM(ok(X)) -> FROM(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FROM(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
             ↳Dependency Graph
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

AND(ok(X1), ok(X2)) -> AND(X1, X2)
AND(mark(X1), X2) -> AND(X1, X2)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

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 19
             ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(AND(x1, x2))=  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 19
             ↳Polo
             ...
               →DP Problem 20
                 ↳Dependency Graph
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  0  
  POL(ok(x1))=  1 + x1  
  POL(IF(x1, x2, x3))=  x2  

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 21
             ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

IF(mark(X1), X2, X3) -> IF(X1, X2, X3)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

IF(mark(X1), X2, X3) -> IF(X1, X2, X3)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(IF(x1, x2, x3))=  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 21
             ↳Polo
             ...
               →DP Problem 22
                 ↳Dependency Graph
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(and(X1, X2)) -> ACTIVE(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(and(x1, x2))=  1 + x1  
  POL(if(x1, x2, x3))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(add(x1, x2))=  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 23
             ↳Polynomial Ordering
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(if(x1, x2, x3))=  x1  
  POL(first(x1, x2))=  1 + x1 + x2  
  POL(add(x1, x2))=  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 23
             ↳Polo
             ...
               →DP Problem 24
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pairs:

ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(if(x1, x2, x3))=  x1  
  POL(add(x1, x2))=  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 23
             ↳Polo
             ...
               →DP Problem 25
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:

ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(if(x1, x2, x3))=  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 23
             ↳Polo
             ...
               →DP Problem 26
                 ↳Dependency Graph
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(and(x1, x2))=  x1 + x2  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  1 + x1  
  POL(add(x1, x2))=  x1 + x2  

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 9
         ↳Polo
           →DP Problem 27
             ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(and(x1, x2))=  x1 + x2  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(add(x1, x2))=  1 + x1 + x2  

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 9
         ↳Polo
           →DP Problem 27
             ↳Polo
             ...
               →DP Problem 28
                 ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(and(x1, x2))=  x1 + x2  
  POL(if(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  

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 9
         ↳Polo
           →DP Problem 27
             ↳Polo
             ...
               →DP Problem 29
                 ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

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 9
         ↳Polo
           →DP Problem 27
             ↳Polo
             ...
               →DP Problem 30
                 ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(from(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

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

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 9
         ↳Polo
           →DP Problem 27
             ↳Polo
             ...
               →DP Problem 31
                 ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 that need to be oriented.

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

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 9
         ↳Polo
           →DP Problem 27
             ↳Polo
             ...
               →DP Problem 32
                 ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo


Dependency Pairs:

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


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  1 + x1 + x2  
  POL(PROPER(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 9
         ↳Polo
           →DP Problem 27
             ↳Polo
             ...
               →DP Problem 33
                 ↳Dependency Graph
       →DP Problem 10
         ↳Polo


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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
         ↳Polo
       →DP Problem 10
         ↳Polynomial Ordering


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

TOP(mark(X)) -> TOP(proper(X))


Additionally, the following usable rules for innermost can be oriented:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
from(ok(X)) -> ok(from(X))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  1  
  POL(proper(x1))=  x1  
  POL(and(x1, x2))=  1 + x1 + x2  
  POL(false)=  1  
  POL(true)=  1  
  POL(mark(x1))=  1 + x1  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(add(x1, x2))=  1 + x1 + x2  
  POL(active(x1))=  x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(0)=  1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  0  
  POL(nil)=  0  
  POL(s(x1))=  1  

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 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 34
             ↳Polynomial Ordering


Dependency Pair:

TOP(ok(X)) -> TOP(active(X))


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

TOP(ok(X)) -> TOP(active(X))


Additionally, the following usable rules for innermost can be oriented:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
from(ok(X)) -> ok(from(X))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(and(x1, x2))=  x2  
  POL(false)=  0  
  POL(true)=  0  
  POL(mark(x1))=  0  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(add(x1, x2))=  x2  
  POL(active(x1))=  x1  
  POL(if(x1, x2, x3))=  x2  
  POL(0)=  0  
  POL(first(x1, x2))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(nil)=  0  
  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 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 34
             ↳Polo
             ...
               →DP Problem 35
                 ↳Dependency Graph


Dependency Pair:


Rules:


active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes