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

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
         ↳Argument Filtering and Ordering


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





The following dependency pairs can be strictly oriented:

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


The following rules can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
or > and > {xor, false}
implies > and > {xor, false}
implies > true
not > true
not > {xor, false}

resulting in one new DP problem.
Used Argument Filtering System:
AND(x1, x2) -> AND(x1, x2)
xor(x1, x2) -> xor(x1, x2)
not(x1) -> not(x1)
or(x1, x2) -> or(x1, x2)
and(x1, x2) -> and(x1, x2)
implies(x1, x2) -> implies(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳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





Using the Dependency Graph resulted in no new DP problems.

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