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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(or(x1, x2))=  1 + x2  
  POL(not(x1))=  0  
  POL(IMPLIES(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pair:

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))





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pair:

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


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

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳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