# Library Coq.Program.Subset

Tactics related to subsets and proof irrelevance.
Require Import Coq.Program.Utils.
Require Import Coq.Program.Equality.
Local Open Scope program_scope.
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 progress f_equal ; apply proof_irrelevance.
Lemma subset_eq : forall A (P : A -> Prop) (n m : sig P), n = m <-> ` n = ` m.
Definition match_eq (A B : Type) (x : A) (fn : forall (y : A | y = x), B) : B :=
fn (exist _ x eq_refl).
Lemma match_eq_rewrite : forall (A B : Type) (x : A) (fn : forall (y : A | y = x), B)
(y : A | y = x),
match_eq A B x fn = fn y.
Now we make a tactic to be able to rewrite a term t which is applied to a match_eq using an arbitrary equality t = u, and u is now the subject of the match.
Ltac rewrite_match_eq H :=
match goal with
[ |- ?T ] =>
match T with
context [ match_eq ?A ?B ?t ?f ] =>
rewrite (match_eq_rewrite A B t f (exist _ _ (eq_sym H)))
end
end.
Otherwise we can simply unfold match_eq and the term trivially reduces to the original definition.
Ltac simpl_match_eq := unfold match_eq ; simpl.