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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

NOT(x) -> XOR(x, true)
OR(x, y) -> XOR(and(x, y), xor(x, y))
OR(x, y) -> AND(x, y)
OR(x, y) -> XOR(x, y)
IMPLIES(x, y) -> XOR(and(x, y), xor(x, true))
IMPLIES(x, y) -> AND(x, y)
IMPLIES(x, y) -> XOR(x, true)
AND(xor(x, y), z) -> XOR(and(x, z), and(y, z))
AND(xor(x, y), z) -> AND(x, z)
AND(xor(x, y), z) -> AND(y, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

AND(xor(x, y), z) -> AND(y, z)
AND(xor(x, y), z) -> AND(x, z)


Rules:


not(x) -> xor(x, true)
or(x, y) -> xor(and(x, y), xor(x, y))
implies(x, y) -> xor(and(x, y), xor(x, true))
and(x, true) -> x
and(x, false) -> false
and(x, x) -> x
and(xor(x, y), z) -> xor(and(x, z), and(y, z))
xor(x, false) -> x
xor(x, x) -> false


Strategy:

innermost




The following dependency pairs can be strictly oriented:

AND(xor(x, y), z) -> AND(y, z)
AND(xor(x, y), z) -> AND(x, z)


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(xor(x1, x2))=  1 + x1 + x2  
  POL(AND(x1, x2))=  x1  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


not(x) -> xor(x, true)
or(x, y) -> xor(and(x, y), xor(x, y))
implies(x, y) -> xor(and(x, y), xor(x, true))
and(x, true) -> x
and(x, false) -> false
and(x, x) -> x
and(xor(x, y), z) -> xor(and(x, z), and(y, z))
xor(x, false) -> x
xor(x, x) -> false


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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