Term Rewriting System R:
[x, y]
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))
DFIB(s(s(x)), y) -> DFIB(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

DFIB(s(s(x)), y) -> DFIB(x, y)
DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))


Rule:


dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

DFIB(s(s(x)), y) -> DFIB(x, y)
DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(dfib(x1, x2))=  0  
  POL(DFIB(x1, x2))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rule:


dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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