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Set Implicit Arguments.
Notation "'fun' { x : A | P } => Q" :=
(fun x:{x:A|P} => Q)
(at level 200, x ident, right associativity).
Notation "( x & ? )" := (@exist _ _ x _) : core_scope.
Definition ex_pi1 (A : Prop) (P : A -> Prop) (t : ex P) : A.
intros.
induction t.
exact x.
Defined.
Lemma ex_pi2 : forall (A : Prop) (P : A -> Prop) (t : ex P),
P (ex_pi1 t).
intros A P.
dependent inversion t.
simpl.
exact p.
Defined.
Notation "` t" := (proj1_sig t) (at level 100) : core_scope.
Notation "'forall' { x : A | P } , Q" :=
(forall x:{x:A|P}, Q)
(at level 200, x ident, right associativity).
Lemma subset_simpl : forall (A : Set) (P : A -> Prop)
(t : sig P), P (` t).
Proof.
intros.
induction t.
simpl ; auto.
Qed.
Ltac destruct_one_pair :=
match goal with
| [H : (ex _) |- _] => destruct H
| [H : (ex2 _) |- _] => destruct H
| [H : (sig _) |- _] => destruct H
| [H : (_ /\ _) |- _] => destruct H
end.
Ltac destruct_exists := repeat (destruct_one_pair) .
Extraction Inline proj1_sig.
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