Section fparts_in_def.
Variable E : Setoid.
(* Add_part. *)
Definition add_part (A : part_set E) (x : E) := union A (single x).
unfold add_part in |- *; auto with algebra.
Qed.
Hint Resolve add_part_comp: algebra.
simpl in |- *; auto with algebra.
Qed.
Hint Resolve add_part_in: algebra.
unfold add_part in |- *.
intros.
apply Trans with (union A (union (single x) (single y))); auto with algebra.
apply Trans with (union A (union (single y) (single x))); auto with algebra.
Qed.
Hint Immediate add_part_com: algebra.
intro A.
case A.
simpl in |- *; unfold pred_compatible, eq_part, in_part in |- *;
simpl in |- *.
intuition eauto.
Qed.
Hint Resolve add_in: algebra.
simpl in |- *.
unfold eq_part, add_part, union, single in |- *; simpl in |- *.
intro.
case A; simpl in |- *.
intros a pa.
intuition.
Qed.
simpl in |- *.
unfold eq_part, add_part, union, single in |- *; simpl in |- *.
intro.
case A; simpl in |- *.
intros a pa.
intuition.
Qed.
intros A B.
case A; case B; simpl in |- *.
intros a pa b pb.
unfold eq_part, add_part, union, single in |- *; simpl in |- *.
intros.
elim (classic (Equal x x0)); intros.
split; intros.
apply pa with x; auto with algebra.
cut (b x).
intro.
absurd (b x); auto with algebra.
apply pb with x0; auto with algebra.
cut (a x).
intro.
absurd (a x); auto with algebra.
apply pa with x0; auto with algebra.
elim (H1 x0); intros.
split; intros.
lapply H3.
intros.
elim H6; intros.
auto with algebra.
absurd (Equal x0 x); auto with algebra.
left; auto with algebra.
lapply H4; intros.
elim H6; intros.
auto with algebra.
absurd (Equal x0 x); auto with algebra.
left.
auto with algebra.
Qed.
(* Minus_part. *)
Definition minus_part (A : part_set E) (x : E) := diff A (single x).
unfold minus_part in |- *; auto with algebra.
Qed.
Hint Resolve minus_part_comp: algebra.
simpl in |- *.
intro.
case A; simpl in |- *.
intros a pa.
intuition.
Qed.
Hint Resolve minus_part_not_in: algebra.
unfold minus_part in |- *.
intros.
simpl in |- *.
unfold eq_part, diff, single in |- *; simpl in |- *.
intro.
intuition.
Qed.
Hint Immediate minus_part_com: algebra.
simpl in |- *.
unfold eq_part, minus_part, diff, single in |- *; simpl in |- *.
intro A.
case A; unfold pred_compatible in |- *; simpl in |- *.
intros a pa x neg_a_x x0.
generalize (pa x0 x).
intuition auto with algebra.
Qed.
Hint Resolve minus_not_in: algebra.
simpl in |- *.
unfold eq_part, minus_part, diff, single in |- *; simpl in |- *.
intro.
case A; simpl in |- *.
intros a pa.
intuition.
Qed.
Hint Resolve minus_trans_not_in: algebra.
(* Some lemmas. *)
simpl in |- *.
unfold eq_part, union, empty in |- *; simpl in |- *.
intuition.
Qed.
Hint Resolve union_unit_l: algebra.
unfold add_part in |- *.
auto with algebra.
Qed.
Hint Resolve single_add: algebra.
simpl in |- *.
unfold eq_part, add_part, minus_part, union, diff, single in |- *;
simpl in |- *.
intro.
case A; simpl in |- *.
intros a pa.
intros.
intuition.
apply pa with x; auto with algebra.
apply NNPP; intuition.
Qed.
Hint Resolve minus_add: algebra.

Lemma add_minus :
forall (A : part_set E) (x : E),
~ in_part x A -> Equal (minus_part (add_part A x) x) A.