The Proof Rules of IPC ====================== /\ intro: if F |- f1 and F |- f2 then F |- f1 /\ f2 /\ elim L: if F |- f1 /\ f2 then F |- f1 /\ elim R: if F |- f1 /\ f2 then F |- f2 => elim: if F |- f and F |- f => g then F |- g => intro: if F,f |- g then F |- f => g assump: f |- f weak: if F |- f then F,g |- f set assump: F, f |- f \/ intro L: if F |- f1 then F |- f1 \/ f2 \/ intro R: if F |- f2 then F |- f1 \/ f2 \/ elim: if F |- f1 \/ f2 and F |- f1 => g and F |- f2 => g then F |- g true intro: F |- true false elim: if F |- false then F |- f ~ intro: if F |- f => false then F |- ~f ~ elim: if F |- ~f then F |- f => false The Proof Rules of IQC ====================== all the above rules for IPC, plus the following: forall intro: if F |- f(x) and x not in FV(F) then F |- forall x, f(x) forall elim: if F |- forall x, f(x) then F |- f(t) exists intro: if F |- f(t) then F |- exists x, f(x) exists elim: if F |- exists x, f(x) and F |- f(x) => g and x not in FV(F,g) then F |- g Classical Logics ================ CPC = IPC + excluded middle CQC = IQC + excluded middle excluded middle: F |- f \/ ~f Theory of Equality ================== eq-refl: F |- t=t eq-symm: if F |- t1=t2 then F |- t2=t1 eq-trans: if F |- t1=t2 and F |- t2=t3 then F |- t1=t3 eq-fun: if F |- t1=u1 and ... and F |- tn=un then fun(t1,...,tn) = fun(u1,...,un) eq-rel: if F |- t1=u1 and ... and F |- tn=un then R(t1,...,tn) = R(u1,...,un)