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
         ↳Usable Rules (Innermost)


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




As we are in the innermost case, we can delete all 9 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Size-Change Principle


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
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:
xor(x1, x2) -> xor(x1, x2)

We obtain no new DP problems.

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