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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

IMPLIES(false, y) -> NOT(false)
IMPLIES(x, false) -> NOT(x)
IMPLIES(not(x), not(y)) -> IMPLIES(y, and(x, y))
IMPLIES(not(x), not(y)) -> AND(x, y)

Furthermore, R contains one SCC. 


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


Dependency Pair:

IMPLIES(not(x), not(y)) -> IMPLIES(y, and(x, y))


Rules:


and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))





The following dependency pair can be strictly oriented:

IMPLIES(not(x), not(y)) -> IMPLIES(y, and(x, y))


The following rules can be oriented:

and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))


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

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


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


Dependency Pair:


Rules:


and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))





Using the Dependency Graph resulted in no new DP problems.

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