Term Rewriting System R:
[x, y, z]
implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))
IMPLIES(x, or(y, z)) -> IMPLIES(x, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)
IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))


Rules:


implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))





The following dependency pair can be strictly oriented:

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))


The following rules can be oriented:

implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(or(x1, x2))=  x1 + x2  
  POL(implies(x1, x2))=  x1 + x2  
  POL(not(x1))=  1 + x1  
  POL(IMPLIES(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
IMPLIES(x1, x2) -> IMPLIES(x1, x2)
not(x1) -> not(x1)
or(x1, x2) -> or(x1, x2)
implies(x1, x2) -> implies(x1, x2)


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


Dependency Pair:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)


Rules:


implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))





The following dependency pair can be strictly oriented:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)


The following rules can be oriented:

implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(or(x1, x2))=  1 + x1 + x2  
  POL(implies(x1, x2))=  x1 + x2  
  POL(not(x1))=  1 + x1  
  POL(IMPLIES(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
IMPLIES(x1, x2) -> IMPLIES(x1, x2)
or(x1, x2) -> or(x1, x2)
implies(x1, x2) -> implies(x1, x2)
not(x1) -> not(x1)


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


Dependency Pair:


Rules:


implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))





Using the Dependency Graph resulted in no new DP problems.

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