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`
`         ↳Polynomial 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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(xor(x1, x2)) =  1 + x1 + x2 POL(AND(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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