Term Rewriting System R:
[x]
half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

HALF(s(s(x))) -> HALF(x)
LOG(s(s(x))) -> LOG(s(half(x)))
LOG(s(s(x))) -> HALF(x)

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS


Dependency Pair:

HALF(s(s(x))) -> HALF(x)


Rules:


half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))





The following dependency pair can be strictly oriented:

HALF(s(s(x))) -> HALF(x)


The following rules can be oriented:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(HALF(x1))=  1 + x1  
  POL(log(x1))=  x1  
  POL(s(x1))=  1 + x1  
  POL(half(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
HALF(x1) -> HALF(x1)
s(x1) -> s(x1)
half(x1) -> half(x1)
log(x1) -> log(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 3
Dependency Graph
       →DP Problem 2
AFS


Dependency Pair:


Rules:


half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Argument Filtering and Ordering


Dependency Pair:

LOG(s(s(x))) -> LOG(s(half(x)))


Rules:


half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))





The following dependency pair can be strictly oriented:

LOG(s(s(x))) -> LOG(s(half(x)))


The following rules can be oriented:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(log(x1))=  x1  
  POL(s(x1))=  1 + x1  
  POL(half(x1))=  x1  
  POL(LOG(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
LOG(x1) -> LOG(x1)
s(x1) -> s(x1)
half(x1) -> half(x1)
log(x1) -> log(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 4
Dependency Graph


Dependency Pair:


Rules:


half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes