Term Rewriting System R:
[x, y]
and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

IMPLIES(false, y) -> NOT(false)
IMPLIES(x, false) -> NOT(x)
IMPLIES(not(x), not(y)) -> IMPLIES(y, and(x, y))
IMPLIES(not(x), not(y)) -> AND(x, y)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pair:

IMPLIES(not(x), not(y)) -> IMPLIES(y, and(x, y))

Rules:

and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))

The following dependency pair can be strictly oriented:

IMPLIES(not(x), not(y)) -> IMPLIES(y, and(x, y))

Additionally, the following usable rules using the Ce-refinement can be oriented:

and(x, false) -> false
and(x, not(false)) -> x

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(and(x1, x2)) =  x1 POL(false) =  0 POL(not(x1)) =  1 + x1 POL(IMPLIES(x1, x2)) =  x1 + x2

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Dependency Graph

Dependency Pair:

Rules:

and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))

Using the Dependency Graph resulted in no new DP problems.

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