(** The following tactics implement a poor-man's solution for proof-irrelevance: it tries to
   factorize every proof of the same proposition in a goal so that equality of such proofs becomes trivial. *)

Ltac on_subset_proof_aux tac T :=
  match T with
    | context [ exist ?P _ ?p ] => try on_subset_proof_aux tac P ; tac p
  end.Ltac on_subset_proof tac :=
  match goal with
    [ |- ?T ] => on_subset_proof_aux tac T
  end.Ltac abstract_any_hyp H' p :=
  match type of p with
    ?X =>
    match goal with
      | [ H : X |- _ ] => fail 1
      | _ => set (H':=p) ; try (change p with H') ; clearbody H'
    end
  end.Ltac abstract_subset_proof :=
  on_subset_proof ltac:(fun p => let H := fresh "eqH" in abstract_any_hyp H p ; simpl in H).Ltac abstract_subset_proofs := repeat abstract_subset_proof.Ltac pi_subset_proof_hyp p :=
  match type of p with
    ?X =>
    match goal with
      | [ H : X |- _ ] =>
        match p with
          | H => fail 2
          | _ => rewrite (proof_irrelevance X p H)
        end
      | _ => fail " No hypothesis with same type "
    end
  end.Ltac pi_subset_proof := on_subset_proof pi_subset_proof_hyp.Ltac pi_subset_proofs := repeat pi_subset_proof.(** The two preceding tactics in sequence. *)

Ltac clear_subset_proofs :=
  abstract_subset_proofs ; simpl in * |- ; pi_subset_proofs ; clear_dups.Ltac pi := repeat f_equal ; apply proof_irrelevance.(* Somewhat trivial definition, but not unfolded automatically hence we can match on [match_eq ?A ?B ?x ?f]
   in tactics. *)

Definition match_eq (A B : Type) (x : A) (fn : {y : A | y = x} -> B) : B :=
  fn (exist _ x eq_refl).

Lemma match_eq_rewrite : forall (A B : Type) (x : A) (fn : {y : A | y = x} -> B) (y : {y:A | y = x}), match_eq A B x fn = fn y.