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
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

and(x, and(y, y)) -> and(x, y)
or(x, or(y, y)) -> or(x, y)


   R
     ↳RRRI
       →TRS2
         ↳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)

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


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, 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





We number the DPs as follows:
  1. AND(x, or(y, z)) -> AND(x, z)
  2. AND(x, or(y, z)) -> AND(x, y)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
or(x1, x2) -> or(x1, x2)

We obtain no new DP problems.

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