Exercise logic.propositional.dnf
Description
Proposition to DNF
Derivation
((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q)))
⇒ logic.propositional.idempand((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q)))
⇒ logic.propositional.idempand((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q)))
⇒ logic.propositional.idempand((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(p /\ ~q)))
⇒ logic.propositional.idempand((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.idempand((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.notnot((q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.notnot((q /\ ~q /\ p /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.compland((F /\ p /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.falsezeroand(F || (T /\ ~r)) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.falsezeroorT /\ ~r /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || ~r) /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.absorpandT /\ ~r /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.truezeroand~r /\ ((~~q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.notnot~r /\ ((q /\ ~~(~q /\ p) /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.notnot~r /\ ((q /\ ~q /\ p /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.compland~r /\ ((F /\ p /\ ~~(p /\ ~q) /\ ~~(~q /\ p)) || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.falsezeroand~r /\ (F || (~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)))
⇒ logic.propositional.falsezeroor~r /\ ~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)
⇒ logic.propositional.idempand~r /\ ~~(~q /\ p) /\ ~~(p /\ ~q)
⇒ logic.propositional.notnot~r /\ ~q /\ p /\ ~~(p /\ ~q)
⇒ logic.propositional.notnot~r /\ ~q /\ p /\ p /\ ~q
⇒ logic.propositional.idempand~r /\ ~q /\ p /\ ~q