mirror of https://github.com/nmvdw/HITs-Examples
Added separation as operation
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Niels
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@ -102,4 +102,59 @@ Section operations.
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* intros p.
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split ; apply p.
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Defined.
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Definition map (A B : Type) (f : A -> B) : FSet A -> FSet B.
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Proof.
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hrecursion.
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- apply ∅.
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- intro a.
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apply {|f a|}.
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- apply (∪).
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- apply assoc.
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- apply comm.
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- apply nl.
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- apply nr.
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- intros.
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apply idem.
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Defined.
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Local Ltac remove_transport
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:= intros ; apply path_forall ; intro Z ; rewrite transport_arrow
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; rewrite transport_const ; simpl.
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Local Ltac pointwise
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:= repeat (f_ap) ; try (apply path_forall ; intro Z2) ;
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rewrite transport_arrow, transport_const, transport_sigma
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; f_ap ; hott_simpl ; simple refine (path_sigma _ _ _ _ _)
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; try (apply transport_const) ; try (apply path_ishprop).
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Lemma separation (A B : Type) : forall (X : FSet A) (f : {a | a ∈ X} -> B), FSet B.
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Proof.
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hinduction.
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- intros f.
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apply ∅.
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- intros a f.
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apply {|f (a; tr idpath)|}.
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- intros X1 X2 HX1 HX2 f.
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pose (fX1 := fun Z : {a : A & a ∈ X1} => f (pr1 Z;tr (inl (pr2 Z)))).
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pose (fX2 := fun Z : {a : A & a ∈ X2} => f (pr1 Z;tr (inr (pr2 Z)))).
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specialize (HX1 fX1).
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specialize (HX2 fX2).
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apply (HX1 ∪ HX2).
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- remove_transport.
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rewrite assoc.
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pointwise.
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- remove_transport.
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rewrite comm.
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pointwise.
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- remove_transport.
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rewrite nl.
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pointwise.
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- remove_transport.
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rewrite nr.
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pointwise.
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- remove_transport.
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rewrite <- (idem (Z (x; tr 1%path))).
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pointwise.
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Defined.
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End operations.
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@ -203,27 +203,19 @@ Section properties.
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apply (tr (inl Xa)).
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Defined.
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(*
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Lemma separation : forall (X : FSet A) (f : {a | a ∈ X} -> B),
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hexists (fun Y : FSet B => forall (b : B),
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b ∈ Y = hexists (fun a => hexists (fun (p : a ∈ X) => f (a;p) = b))).
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Lemma separation_isIn (B : Type) : forall (X : FSet A) (f : {a | a ∈ X} -> B) (b : B),
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b ∈ (separation A B X f) = hexists (fun a : A => hexists (fun (p : a ∈ X) => f (a;p) = b)).
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Proof.
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hinduction ; try (intros ; apply path_forall ; intro ; apply path_ishprop).
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- intros ; simpl.
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apply tr.
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exists ∅.
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intros ; simpl.
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apply path_iff_hprop ; try contradiction.
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intros.
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intros X.
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strip_truncations.
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destruct X as [a X].
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strip_truncations.
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destruct X as [p X].
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assumption.
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- intros a f.
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apply tr.
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exists {|f (a;tr idpath)|}.
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intros.
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- intros.
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apply path_iff_hprop ; simpl.
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* intros ; strip_truncations.
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apply tr.
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@ -231,7 +223,7 @@ Section properties.
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apply tr.
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exists (tr idpath).
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apply X^.
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* intros ; strip_truncations.
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* intros X ; strip_truncations.
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destruct X as [a0 X].
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strip_truncations.
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destruct X as [X q].
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@ -243,26 +235,19 @@ Section properties.
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simple refine (path_sigma _ _ _ _ _).
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** apply p.
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** apply path_ishprop.
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- intros X1 X2 HX1 HX2 f.
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- intros X1 X2 HX1 HX2 f b.
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pose (fX1 := fun Z : {a : A & a ∈ X1} => f (pr1 Z;tr (inl (pr2 Z)))).
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pose (fX2 := fun Z : {a : A & a ∈ X2} => f (pr1 Z;tr (inr (pr2 Z)))).
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specialize (HX1 fX1).
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specialize (HX2 fX2).
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strip_truncations.
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destruct HX1 as [Y1 fY1].
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destruct HX2 as [Y2 fY2].
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apply tr.
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exists (Y1 ∪ Y2).
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intros b.
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specialize (fY1 b).
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specialize (fY2 b).
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cbn.
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rewrite fY1, fY2.
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specialize (HX1 fX1 b).
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specialize (HX2 fX2 b).
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apply path_iff_hprop.
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* intros.
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* intros X.
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rewrite union_isIn in X.
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strip_truncations.
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destruct X as [X | X] ; strip_truncations.
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** destruct X as [a Ha].
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destruct X as [X | X].
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** rewrite HX1 in X.
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strip_truncations.
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destruct X as [a Ha].
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apply tr.
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exists a.
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strip_truncations.
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@ -270,9 +255,10 @@ Section properties.
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apply tr.
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exists (tr(inl p)).
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rewrite <- pa.
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unfold fX1.
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reflexivity.
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** destruct X as [a Ha].
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** rewrite HX2 in X.
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strip_truncations.
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destruct X as [a Ha].
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apply tr.
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exists a.
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strip_truncations.
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@ -280,15 +266,17 @@ Section properties.
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apply tr.
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exists (tr(inr p)).
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rewrite <- pa.
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unfold fX2.
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reflexivity.
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* intros.
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strip_truncations.
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destruct X as [a X].
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strip_truncations.
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destruct X as [Ha p].
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rewrite union_isIn.
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simple refine (Trunc_ind _ _ Ha) ; intros [Ha1 | Ha2].
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** refine (tr(inl(tr _))).
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** refine (tr(inl _)).
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rewrite HX1.
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apply tr.
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exists a.
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apply tr.
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exists Ha1.
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@ -296,7 +284,9 @@ Section properties.
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unfold fX1.
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repeat f_ap.
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apply path_ishprop.
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** refine (tr(inr(tr _))).
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** refine (tr(inr _)).
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rewrite HX2.
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apply tr.
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exists a.
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apply tr.
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exists Ha2.
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@ -305,6 +295,5 @@ Section properties.
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repeat f_ap.
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apply path_ishprop.
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Defined.
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*)
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End properties.
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