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Lattice.v
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Inductive label : Type :=
| High
| Low.
Definition join l1 l2 :=
match l1, l2 with
| High, High => High
| High, Low => High
| Low, High => High
| Low, Low => Low
end.
Lemma join_idempotent :
forall l,
l = join l l.
Proof.
destruct l; reflexivity.
Qed.
Definition flows_to l1 l2 :=
l2 = join l1 l2.
Lemma flows_to_refl :
forall l,
flows_to l l.
Proof.
unfold flows_to.
apply join_idempotent.
Qed.
Lemma flows_to_trans :
forall l l' l'',
flows_to l l' ->
flows_to l' l'' ->
flows_to l l''.
Proof.
intros l l' l'';
destruct l;
destruct l';
destruct l'';
auto.
intros.
apply flows_to_refl.
Qed.
Lemma flows_to_antisym :
forall l l',
flows_to l l' ->
flows_to l' l ->
l = l'.
Proof.
destruct l;
destruct l';
intros;
auto.
Qed.
Lemma high_is_top :
forall l,
flows_to l High.
Proof.
destruct l; reflexivity.
Qed.
Lemma low_is_bot :
forall l,
flows_to Low l.
Proof.
destruct l; reflexivity.
Qed.
Lemma join_high_r :
forall l,
join l High = High.
Proof.
destruct l; reflexivity.
Qed.
Lemma join_high_l :
forall l,
join High l = High.
Proof.
destruct l; reflexivity.
Qed.
Lemma join_low_r :
forall l,
join l Low = l.
Proof.
destruct l; reflexivity.
Qed.
Lemma join_low_l :
forall l,
join Low l = l.
Proof.
destruct l; reflexivity.
Qed.
Lemma join_is_upper_bound :
forall l l',
flows_to l (join l l') /\ flows_to l' (join l l').
Proof.
destruct l; destruct l'; split; (auto ) || (unfold flows_to; reflexivity).
Qed.
Lemma join_is_least_upper_bound :
forall l l' u,
flows_to l u ->
flows_to l' u ->
flows_to (join l l') u.
Proof.
destruct l; destruct l'; destruct u; intros; auto.
Qed.
Lemma flows_to_questionable :
forall l l' l'',
flows_to l l' ->
flows_to l l'' ->
flows_to l (join l'' l').
Proof.
destruct l; destruct l'; destruct l''; intros; try reflexivity.
inversion H0.
Qed.
Lemma join_comm :
forall l l',
join l l' = join l' l.
Proof.
intros.
destruct (join_is_upper_bound l l').
destruct (join_is_upper_bound l' l).
assert (flows_to (join l l') (join l' l)).
apply join_is_least_upper_bound; assumption.
assert (flows_to (join l' l) (join l l')).
apply join_is_least_upper_bound; assumption.
apply flows_to_antisym; auto.
Qed.
Lemma join_assoc :
forall l l' l'',
join (join l l') l'' = join l (join l' l'').
Proof.
destruct l; destruct l'; destruct l''; reflexivity.
Lemma join_flows_to_join :
forall l l' ls ls',
flows_to l l' ->
flows_to ls ls' ->
flows_to (join l ls) (join l' ls').
Proof.
intros l l' ls ls' l_l'_flows_to ls_ls'_flows_to.
assert (flows_to l' (join l' ls')).
apply join_is_upper_bound.
assert (flows_to l (join l' ls')).
apply (flows_to_trans l l' (join l' ls')).
apply l_l'_flows_to.
apply H.
clear H.
assert (flows_to ls' (join l' ls')).
apply join_is_upper_bound.
assert (flows_to ls (join l' ls')).
apply (flows_to_trans ls ls' (join l' ls')).
apply ls_ls'_flows_to.
apply H.
clear H.
apply join_is_least_upper_bound.
apply H0.
apply H1.
Qed.