Term Rewriting System R:
[x, y]
minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

MINUS(+(x, y)) -> MINUS(minus(minus(x)))
MINUS(+(x, y)) -> MINUS(minus(x))
MINUS(+(x, y)) -> MINUS(x)
MINUS(+(x, y)) -> MINUS(minus(minus(y)))
MINUS(+(x, y)) -> MINUS(minus(y))
MINUS(+(x, y)) -> MINUS(y)
MINUS(*(x, y)) -> MINUS(minus(minus(x)))
MINUS(*(x, y)) -> MINUS(minus(x))
MINUS(*(x, y)) -> MINUS(x)
MINUS(*(x, y)) -> MINUS(minus(minus(y)))
MINUS(*(x, y)) -> MINUS(minus(y))
MINUS(*(x, y)) -> MINUS(y)
F(minus(x)) -> MINUS(minus(minus(f(x))))
F(minus(x)) -> MINUS(minus(f(x)))
F(minus(x)) -> MINUS(f(x))
F(minus(x)) -> F(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

MINUS(*(x, y)) -> MINUS(y)
MINUS(*(x, y)) -> MINUS(minus(y))
MINUS(*(x, y)) -> MINUS(minus(minus(y)))
MINUS(*(x, y)) -> MINUS(x)
MINUS(*(x, y)) -> MINUS(minus(x))
MINUS(*(x, y)) -> MINUS(minus(minus(x)))
MINUS(+(x, y)) -> MINUS(y)
MINUS(+(x, y)) -> MINUS(minus(y))
MINUS(+(x, y)) -> MINUS(minus(minus(y)))
MINUS(+(x, y)) -> MINUS(x)
MINUS(+(x, y)) -> MINUS(minus(x))
MINUS(+(x, y)) -> MINUS(minus(minus(x)))


Rules:


minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MINUS(*(x, y)) -> MINUS(y)
MINUS(*(x, y)) -> MINUS(minus(y))
MINUS(*(x, y)) -> MINUS(minus(minus(y)))
MINUS(*(x, y)) -> MINUS(x)
MINUS(*(x, y)) -> MINUS(minus(x))
MINUS(*(x, y)) -> MINUS(minus(minus(x)))
MINUS(+(x, y)) -> MINUS(y)
MINUS(+(x, y)) -> MINUS(minus(y))
MINUS(+(x, y)) -> MINUS(minus(minus(y)))
MINUS(+(x, y)) -> MINUS(x)
MINUS(+(x, y)) -> MINUS(minus(x))
MINUS(+(x, y)) -> MINUS(minus(minus(x)))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(minus(x1))=  x1  
  POL(*(x1, x2))=  1 + x1 + x2  
  POL(MINUS(x1))=  1 + x1  
  POL(+(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes