header {* Classical sequent calculus: examples with propositional connectives *}
theory Propositional
imports LK
begin
text "absorptive laws of & and | "
lemma "|- P & P <-> P"
by fast_prop
lemma "|- P | P <-> P"
by fast_prop
text "commutative laws of & and | "
lemma "|- P & Q <-> Q & P"
by fast_prop
lemma "|- P | Q <-> Q | P"
by fast_prop
text "associative laws of & and | "
lemma "|- (P & Q) & R <-> P & (Q & R)"
by fast_prop
lemma "|- (P | Q) | R <-> P | (Q | R)"
by fast_prop
text "distributive laws of & and | "
lemma "|- (P & Q) | R <-> (P | R) & (Q | R)"
by fast_prop
lemma "|- (P | Q) & R <-> (P & R) | (Q & R)"
by fast_prop
text "Laws involving implication"
lemma "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"
by fast_prop
lemma "|- (P & Q --> R) <-> (P--> (Q-->R))"
by fast_prop
lemma "|- (P --> Q & R) <-> (P-->Q) & (P-->R)"
by fast_prop
text "Classical theorems"
lemma "|- P|Q --> P| ~P&Q"
by fast_prop
lemma "|- (P-->Q)&(~P-->R) --> (P&Q | R)"
by fast_prop
lemma "|- P&Q | ~P&R <-> (P-->Q)&(~P-->R)"
by fast_prop
lemma "|- (P-->Q) | (P-->R) <-> (P --> Q | R)"
by fast_prop
lemma "|- (P<->Q) <-> (Q<->P)"
by fast_prop
lemma "|- ~ (P <-> ~P)"
by fast_prop
lemma "|- (P-->Q) <-> (~Q --> ~P)"
by fast_prop
lemma "|- ~ ~ P <-> P"
by fast_prop
lemma "|- ~(P-->Q) --> (Q-->P)"
by fast_prop
lemma "|- (~P-->Q) <-> (~Q --> P)"
by fast_prop
lemma "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))"
by fast_prop
lemma "|- P | ~ P"
by fast_prop
lemma "|- P | ~ ~ ~ P"
by fast_prop
lemma "|- ((P-->Q) --> P) --> P"
by fast_prop
lemma "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
by fast_prop
lemma "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q"
by fast_prop
lemma "|- P<->P"
by fast_prop
lemma "|- ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"
by fast_prop
lemma "|- P | (Q & R) <-> (P | Q) & (P | R)"
by fast_prop
lemma "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))"
by fast_prop
lemma "|- (P --> Q) <-> (~P | Q)"
by fast_prop
lemma "|- (P-->Q) | (Q-->P)"
by fast_prop
lemma "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
by fast_prop
end