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
         ↳Forward Instantiation Transformation


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




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

AND(xor(x, y), z) -> AND(x, z)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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




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

AND(xor(x, y), z) -> AND(y, z)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pairs:

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


The following usable rules for innermost can be oriented:

xor(x, false) -> x
xor(x, x) -> false


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(xor(x1, x2))=  1 + x1 + x2  
  POL(false)=  0  
  POL(AND(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
AND(x1, x2) -> AND(x1, x2)
xor(x1, x2) -> xor(x1, x2)


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