Exercise logic.propositional.consequence

Description
Prove that formula is a logical consequence of a set of formulas

Examples

easy (5)

p /\ ~p => q

p -> (p -> q), p => q

p -> q, p => q

p || (q /\ r) => ~p -> r

p -> q, ~q => ~p

medium (8)

p -> q, p -> ~q => ~p

~p -> ~q, q => p

p -> q, p -> r => p -> (q /\ r)

p -> q, q -> r => p -> r

p -> (q -> r), q => p -> r

q => p -> (p /\ q)

~q => (p || q) -> p

~q, p || q => p

difficult (11)

p -> q, r -> s => (p /\ r) -> (q /\ s)

~r <-> q => p -> (q || r)

p -> (q /\ r) => (p /\ q) <-> (p /\ r)

p <-> r, q <-> s => (p -> q) <-> (r -> s)

p <-> r, q <-> s => (p /\ q) <-> (r /\ s)

p <-> r, q <-> s => (p || q) <-> (r || s)

q <-> s => (p /\ q) <-> (p /\ s)

q => (p /\ q) <-> p

~q => (p || q) <-> p

p <-> q, p => q

p <-> q, ~p => ~q