Section CommonTheorems.
Context {orig_base_params : BaseParams}.
Context {one_node_params : OneNodeParams orig_base_params}.
Context {raft_params : RaftParams orig_base_params}.
Proof using.
unfold uniqueIndices.
eauto using NoDup_map_elim.
Qed.
Proof using.
intros.
simpl in *. intuition;
find_apply_hyp_hyp; intuition.
Qed.
Proof using.
induction ys; intros; simpl in *.
- break_match; intuition.
- break_match; intuition.
subst. apply sorted_cons; eauto.
intros. eauto using subseq_In.
Qed.
Proof using.
induction l; intros.
- simpl in *. intuition.
- simpl in *. intuition.
+ subst. auto with *.
+ find_apply_hyp_hyp. lia.
Qed.
Proof using.
induction l; intros.
- simpl in *. intuition.
- simpl in *. intuition.
+ subst. auto with *.
+ find_apply_hyp_hyp. lia.
Qed.
Proof using.
intros. intro.
find_apply_lem_hyp maxIndex_is_max; auto.
subst. lia.
Qed.
Proof using.
destruct l; intros; simpl in *; eauto; congruence.
Qed.
Proof using.
induction l; intros; simpl; auto.
repeat break_match; intuition.
- find_inversion. eauto using subseq_refl.
- right. find_reverse_rewrite. auto.
Qed.
Proof using.
intros. eauto using removeAfterIndex_subseq, sorted_subseq.
Qed.
Proof using.
eauto using removeAfterIndex_subseq, subseq_In.
Qed.
Proof using.
induction l; intros; intro.
- intuition.
- simpl in *. break_match; try discriminate. intuition.
+ subst. rewrite <- beq_nat_refl in *. discriminate.
+ find_copy_apply_hyp_hyp. intuition. break_if; do_bool; eauto. lia.
Qed.
Proof using.
induction l; intros.
- discriminate.
- simpl in *. break_match.
+ find_inversion. auto.
+ break_if; eauto; discriminate.
Qed.
Proof using.
induction l; intros.
- discriminate.
- simpl in *. break_match.
+ find_inversion. apply beq_nat_true in Heqb. auto.
+ break_if; eauto; discriminate.
Qed.
Proof using.
eauto using subseq_NoDup, removeAfterIndex_subseq.
Qed.
Notation disjoint xs ys := (forall x, In x xs -> In x ys -> False).
Proof using.
induction xs; intros.
- intuition.
- simpl in *. break_if; simpl in *; intuition.
subst. do_bool. lia.
Qed.
Proof using.
induction xs; intros.
- simpl in *. intuition.
- simpl in *.
break_if; simpl in *; do_bool; intuition; subst; auto.
find_apply_hyp_hyp. lia.
Qed.
Proof using.
induction xs; intros.
- simpl in *. intuition.
- simpl in *.
break_match; simpl in *; intuition;
subst; do_bool;
match goal with
| e : entry, H : forall _, _ = _ \/ _ -> _ |- _ =>
specialize (H e)
end;
intuition.
+ eauto using removeAfterIndex_le_In.
+ find_apply_hyp_hyp. intuition.
match goal with
| _ : eIndex ?e <= ?li, _ : eIndex ?e > eIndex ?e' |- _ =>
assert (eIndex e' <= li) by lia
end.
eauto using removeAfterIndex_le_In.
Qed.
Proof using.
induction xs; intros.
- reflexivity.
- simpl.
find_copy_apply_hyp_hyp.
repeat (break_if; simpl in *; intuition); try discriminate.
do_bool. lia.
Qed.
Proof using.
induction xs; intros; simpl.
- auto.
- repeat (break_match; simpl); intuition; try discriminate.
+ eauto using subseq_refl.
+ do_bool. assert (j < i) by lia.
rewrite removeAfterIndex_le with (j := i) (i := j) at 1; auto; lia.
+ do_bool. assert (i < j) by lia.
rewrite removeAfterIndex_le with (i := i) (j := j) at 2; auto; lia.
Qed.
Proof using.
auto using subseq_subseq_eq, removeAfterIndex_2_subseq.
Qed.
Proof using.
intros.
pose proof Min.min_spec i j. intuition.
- find_rewrite. rewrite removeAfterIndex_le with (i := i) (j := j) at 2;
eauto using removeAfterIndex_comm; lia.
- find_rewrite.
rewrite <- removeAfterIndex_le with (i := j) (j := i);
auto; lia.
Qed.
Proof using.
induction xs; intros; simpl in *; intuition; break_match; try discriminate.
- subst. do_bool. congruence.
- do_bool. break_if; eauto.
do_bool. find_apply_hyp_hyp. intuition.
Qed.
Proof using.
intros.
eapply NoDup_map_elim with (f := eIndex); eauto using findAtIndex_in, removeAfterIndex_in.
apply findAtIndex_index in H0.
apply findAtIndex_index in H1.
congruence.
Qed.
Proof using.
unfold uniqueIndices.
induction ys; intros.
- simpl in *. break_match; intuition.
- simpl in *. break_match; intuition.
+ simpl. constructor.
+ subst. simpl in *. invc H0. constructor; auto.
intro. apply H3.
eapply subseq_In; eauto.
apply subseq_map. auto.
+ subst. invc H0. eauto.
Qed.

Lemma subseq_findGtIndex :
forall xs i,
subseq (findGtIndex xs i) xs.