Term Rewriting System R:
[x, y, z]
and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AND(x, or(y, z)) -> OR(and(x, y), and(x, z))
AND(x, or(y, z)) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)
AND(x, and(y, y)) -> AND(x, y)
OR(x, or(y, y)) -> OR(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)


Strategy:

innermost




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

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

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


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)


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

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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