Term Rewriting System R:
[x]
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

REC(rec(x)) -> SENT(rec(x))
REC(sent(x)) -> SENT(rec(x))
REC(sent(x)) -> REC(x)
REC(no(x)) -> SENT(rec(x))
REC(no(x)) -> REC(x)
REC(bot) -> SENT(bot)
REC(up(x)) -> REC(x)
SENT(up(x)) -> SENT(x)
NO(up(x)) -> NO(x)
TOP(rec(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> CHECK(rec(x))
TOP(rec(up(x))) -> REC(x)
TOP(sent(up(x))) -> TOP(check(rec(x)))
TOP(sent(up(x))) -> CHECK(rec(x))
TOP(sent(up(x))) -> REC(x)
TOP(no(up(x))) -> TOP(check(rec(x)))
TOP(no(up(x))) -> CHECK(rec(x))
TOP(no(up(x))) -> REC(x)
CHECK(up(x)) -> CHECK(x)
CHECK(sent(x)) -> SENT(check(x))
CHECK(sent(x)) -> CHECK(x)
CHECK(rec(x)) -> REC(check(x))
CHECK(rec(x)) -> CHECK(x)
CHECK(no(x)) -> NO(check(x))
CHECK(no(x)) -> CHECK(x)

Furthermore, R contains five SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pair:

SENT(up(x)) -> SENT(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

SENT(up(x)) -> SENT(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(rec(x1))=  1 + x1  
  POL(no(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  1 + x1  
  POL(check(x1))=  x1  
  POL(SENT(x1))=  1 + x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
SENT(x1) -> SENT(x1)
up(x1) -> up(x1)
rec(x1) -> rec(x1)
sent(x1) -> sent(x1)
no(x1) -> no(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 6
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pair:


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pair:

NO(up(x)) -> NO(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

NO(up(x)) -> NO(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(NO(x1))=  1 + x1  
  POL(rec(x1))=  1 + x1  
  POL(no(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  1 + x1  
  POL(check(x1))=  x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
NO(x1) -> NO(x1)
up(x1) -> up(x1)
rec(x1) -> rec(x1)
sent(x1) -> sent(x1)
no(x1) -> no(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 7
             ↳Dependency Graph
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pair:


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Argument Filtering and Ordering
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pairs:

REC(up(x)) -> REC(x)
REC(no(x)) -> REC(x)
REC(sent(x)) -> REC(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

REC(no(x)) -> REC(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(no(x1))=  1 + x1  
  POL(rec(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  x1  
  POL(check(x1))=  x1  
  POL(REC(x1))=  1 + x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
REC(x1) -> REC(x1)
no(x1) -> no(x1)
sent(x1) -> sent(x1)
up(x1) -> up(x1)
rec(x1) -> rec(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
           →DP Problem 8
             ↳Argument Filtering and Ordering
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pairs:

REC(up(x)) -> REC(x)
REC(sent(x)) -> REC(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

REC(sent(x)) -> REC(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(rec(x1))=  1 + x1  
  POL(no(x1))=  1 + x1  
  POL(bot)=  0  
  POL(up(x1))=  x1  
  POL(check(x1))=  x1  
  POL(REC(x1))=  1 + x1  
  POL(sent(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
REC(x1) -> REC(x1)
sent(x1) -> sent(x1)
up(x1) -> up(x1)
rec(x1) -> rec(x1)
no(x1) -> no(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
           →DP Problem 8
             ↳AFS
             ...
               →DP Problem 9
                 ↳Argument Filtering and Ordering
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pair:

REC(up(x)) -> REC(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

REC(up(x)) -> REC(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(rec(x1))=  1 + x1  
  POL(no(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  1 + x1  
  POL(check(x1))=  x1  
  POL(REC(x1))=  1 + x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
REC(x1) -> REC(x1)
up(x1) -> up(x1)
rec(x1) -> rec(x1)
sent(x1) -> sent(x1)
no(x1) -> no(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
           →DP Problem 8
             ↳AFS
             ...
               →DP Problem 10
                 ↳Dependency Graph
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS


Dependency Pair:


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳Argument Filtering and Ordering
       →DP Problem 5
         ↳AFS


Dependency Pairs:

CHECK(no(x)) -> CHECK(x)
CHECK(rec(x)) -> CHECK(x)
CHECK(sent(x)) -> CHECK(x)
CHECK(up(x)) -> CHECK(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pairs can be strictly oriented:

CHECK(rec(x)) -> CHECK(x)
CHECK(up(x)) -> CHECK(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(no(x1))=  x1  
  POL(rec(x1))=  1 + x1  
  POL(bot)=  0  
  POL(up(x1))=  1 + x1  
  POL(check(x1))=  x1  
  POL(CHECK(x1))=  1 + x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
up(x1) -> up(x1)
no(x1) -> no(x1)
sent(x1) -> sent(x1)
rec(x1) -> rec(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
           →DP Problem 11
             ↳Argument Filtering and Ordering
       →DP Problem 5
         ↳AFS


Dependency Pairs:

CHECK(no(x)) -> CHECK(x)
CHECK(sent(x)) -> CHECK(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

CHECK(no(x)) -> CHECK(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  x1  
  POL(no(x1))=  1 + x1  
  POL(rec(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  x1  
  POL(check(x1))=  x1  
  POL(CHECK(x1))=  1 + x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
no(x1) -> no(x1)
sent(x1) -> sent(x1)
rec(x1) -> rec(x1)
up(x1) -> up(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
           →DP Problem 11
             ↳AFS
             ...
               →DP Problem 12
                 ↳Argument Filtering and Ordering
       →DP Problem 5
         ↳AFS


Dependency Pair:

CHECK(sent(x)) -> CHECK(x)


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

CHECK(sent(x)) -> CHECK(x)


The following rules can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(rec(x1))=  1 + x1  
  POL(no(x1))=  1 + x1  
  POL(bot)=  0  
  POL(up(x1))=  x1  
  POL(check(x1))=  x1  
  POL(CHECK(x1))=  1 + x1  
  POL(sent(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
sent(x1) -> sent(x1)
rec(x1) -> rec(x1)
no(x1) -> no(x1)
up(x1) -> up(x1)
top(x1) -> top(x1)
check(x1) -> check(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
           →DP Problem 11
             ↳AFS
             ...
               →DP Problem 13
                 ↳Dependency Graph
       →DP Problem 5
         ↳AFS


Dependency Pair:


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳Argument Filtering and Ordering


Dependency Pairs:

TOP(no(up(x))) -> TOP(check(rec(x)))
TOP(sent(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> TOP(check(rec(x)))


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

TOP(no(up(x))) -> TOP(check(rec(x)))


The following rules can be oriented:

check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(no(x1))=  1 + x1  
  POL(rec(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  x1  
  POL(check(x1))=  x1  
  POL(TOP(x1))=  x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
no(x1) -> no(x1)
check(x1) -> check(x1)
up(x1) -> up(x1)
rec(x1) -> rec(x1)
sent(x1) -> sent(x1)
top(x1) -> top(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS
           →DP Problem 14
             ↳Argument Filtering and Ordering


Dependency Pairs:

TOP(sent(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> TOP(check(rec(x)))


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)





The following dependency pair can be strictly oriented:

TOP(rec(up(x))) -> TOP(check(rec(x)))


The following rules can be oriented:

check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(top(x1))=  1 + x1  
  POL(rec(x1))=  1 + x1  
  POL(no(x1))=  x1  
  POL(bot)=  0  
  POL(up(x1))=  1 + x1  
  POL(check(x1))=  x1  
  POL(TOP(x1))=  1 + x1  
  POL(sent(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
rec(x1) -> rec(x1)
check(x1) -> check(x1)
up(x1) -> up(x1)
sent(x1) -> sent(x1)
no(x1) -> no(x1)
top(x1) -> top(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
       →DP Problem 5
         ↳AFS
           →DP Problem 14
             ↳AFS
             ...
               →DP Problem 15
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

TOP(sent(up(x))) -> TOP(check(rec(x)))


Rules:


rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)




Termination of R could not be shown.
Duration:
0:04 minutes