Term Rewriting System R:
[X, XS, X1, X2]
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

active(tail(cons(X, XS))) -> mark(XS)

where the Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(active(x1))=  x1  
  POL(proper(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  1 + x1  
  POL(zeros)=  0  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  x1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(tail(X)) -> TAIL(active(X))
ACTIVE(tail(X)) -> ACTIVE(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(tail(X)) -> TAIL(proper(X))
PROPER(tail(X)) -> PROPER(X)
TAIL(mark(X)) -> TAIL(X)
TAIL(ok(X)) -> TAIL(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 five SCCs. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳MRR


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
  2. CONS(mark(X1), X2) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳Size-Change Principle
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳MRR


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. TAIL(ok(X)) -> TAIL(X)
  2. TAIL(mark(X)) -> TAIL(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳Size-Change Principle
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳MRR


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(tail(X)) -> ACTIVE(X)
  2. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
tail(x1) -> tail(x1)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳Size-Change Principle
           →DP Problem 5
             ↳MRR


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(tail(X)) -> PROPER(X)
  2. PROPER(cons(X1, X2)) -> PROPER(X2)
  3. PROPER(cons(X1, X2)) -> PROPER(X1)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
tail(x1) -> tail(x1)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳Modular Removal of Rules


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(active(x1))=  x1  
  POL(proper(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(zeros)=  0  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  

We have the following set D of usable symbols: {proper, active, 0, cons, tail, zeros, mark, TOP, ok}
No Dependency Pairs can be deleted.
2 non usable rules have been deleted.

The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳MRR
             ...
               →DP Problem 6
                 ↳Negative Polynomial Order


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)





The following Dependency Pair can be strictly oriented using the given order.

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


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1 + 1

POL( active(x1) ) = x1

POL( mark(x1) ) = x1 + 1

POL( proper(x1) ) = x1 + 1

POL( cons(x1, x2) ) = x1

POL( zeros ) = 1

POL( 0 ) = 0

POL( tail(x1) ) = x1


This results in one new DP problem. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳MRR
             ...
               →DP Problem 7
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(zeros) -> mark(cons(0, zeros))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(zeros)=  0  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  

We have the following set D of usable symbols: {proper, 0, cons, tail, mark, zeros, TOP, ok}
No Dependency Pairs can be deleted.
3 non usable rules have been deleted.

The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳SCP
           →DP Problem 2
             ↳SCP
           →DP Problem 3
             ↳SCP
           →DP Problem 4
             ↳SCP
           →DP Problem 5
             ↳MRR
             ...
               →DP Problem 8
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rules:


proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))





We have the following set of usable rules:

proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(tail(X)) -> tail(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(zeros)=  0  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  

We have the following set D of usable symbols: {proper, 0, cons, tail, mark, zeros, TOP, ok}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.


Termination of R successfully shown.
Duration:
0:02 minutes