Term Rewriting System R:
[x]
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

TOP(free(x)) -> TOP(check(new(x)))
TOP(free(x)) -> CHECK(new(x))
TOP(free(x)) -> NEW(x)
NEW(free(x)) -> NEW(x)
OLD(free(x)) -> OLD(x)
CHECK(free(x)) -> CHECK(x)
CHECK(new(x)) -> NEW(check(x))
CHECK(new(x)) -> CHECK(x)
CHECK(old(x)) -> OLD(check(x))
CHECK(old(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
Remaining


Dependency Pair:

NEW(free(x)) -> NEW(x)


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

NEW(free(x)) -> NEW(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(NEW(x1))=  x1  
  POL(free(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
NEW(x1) -> NEW(x1)
free(x1) -> free(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
Remaining


Dependency Pair:


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




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
Remaining


Dependency Pair:

OLD(free(x)) -> OLD(x)


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

OLD(free(x)) -> OLD(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(free(x1))=  1 + x1  
  POL(OLD(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
OLD(x1) -> OLD(x1)
free(x1) -> free(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
Remaining


Dependency Pair:


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




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
Remaining


Dependency Pair:

CHECK(free(x)) -> CHECK(x)


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

CHECK(free(x)) -> CHECK(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(free(x1))=  1 + x1  
  POL(CHECK(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
free(x1) -> free(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 8
Dependency Graph
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pair:


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




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
Remaining


Dependency Pairs:

CHECK(old(x)) -> CHECK(x)
CHECK(new(x)) -> CHECK(x)


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

CHECK(new(x)) -> CHECK(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(old(x1))=  x1  
  POL(CHECK(x1))=  x1  
  POL(new(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
new(x1) -> new(x1)
old(x1) -> old(x1)


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


Dependency Pair:

CHECK(old(x)) -> CHECK(x)


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

CHECK(old(x)) -> CHECK(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(old(x1))=  1 + x1  
  POL(CHECK(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
old(x1) -> old(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
           →DP Problem 9
AFS
             ...
               →DP Problem 10
Dependency Graph
       →DP Problem 5
Remaining


Dependency Pair:


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost




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
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

TOP(free(x)) -> TOP(check(new(x)))


Rules:


top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)


Strategy:

innermost



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