Term Rewriting System R:
[x, y, z]
xor(x, F) -> x
xor(x, neg(x)) -> F
xor(x, x) -> F
and(x, T) -> x
and(x, F) -> F
and(x, x) -> x
and(xor(x, y), z) -> xor(and(x, z), and(y, z))
impl(x, y) -> xor(and(x, y), xor(x, T))
or(x, y) -> xor(and(x, y), xor(x, y))
equiv(x, y) -> xor(x, xor(y, T))
neg(x) -> xor(x, T)

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

xor(x, neg(x)) -> F


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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)
IMPL(x, y) -> XOR(and(x, y), xor(x, T))
IMPL(x, y) -> AND(x, y)
IMPL(x, y) -> XOR(x, T)
OR(x, y) -> XOR(and(x, y), xor(x, y))
OR(x, y) -> AND(x, y)
OR(x, y) -> XOR(x, y)
EQUIV(x, y) -> XOR(x, xor(y, T))
EQUIV(x, y) -> XOR(y, T)
NEG(x) -> XOR(x, T)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


xor(x, F) -> x
xor(x, x) -> F
and(x, T) -> x
and(x, F) -> F
and(x, x) -> x
and(xor(x, y), z) -> xor(and(x, z), and(y, z))
impl(x, y) -> xor(and(x, y), xor(x, T))
or(x, y) -> xor(and(x, y), xor(x, y))
equiv(x, y) -> xor(x, xor(y, T))
neg(x) -> xor(x, T)





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):
{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:
xor(x1, x2) -> xor(x1, x2)

We obtain no new DP problems.

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