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https://github.com/nmvdw/HITs-Examples
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Merge branch 'ezsplit'
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@@ -5,34 +5,34 @@ Require Import fsets.properties.
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Section finite_hott.
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Variable (A : Type).
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Context `{Univalence} `{IsHSet A}.
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Context `{Univalence}.
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(* A subobject is B-finite if its extension is B-finite as a type *)
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Definition Bfin (X : Sub A) : hProp := BuildhProp (Finite {a : A & a ∈ X}).
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Global Instance singleton_contr a : Contr {b : A & b ∈ {|a|}}.
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Global Instance singleton_contr a `{IsHSet A} : Contr {b : A & b ∈ {|a|}}.
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Proof.
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exists (a; tr idpath).
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intros [b p].
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simple refine (Trunc_ind (fun p => (a; tr 1%path) = (b; p)) _ p).
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clear p; intro p. simpl.
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apply path_sigma' with (p^).
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apply path_ishprop.
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apply path_sigma_hprop; simpl.
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apply p^.
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Defined.
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Definition singleton_fin_equiv' a : Fin 1 -> {b : A & b ∈ {|a|}}.
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Proof.
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intros _. apply (center {b : A & b ∈ {|a|}}).
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intros _. apply (a; tr idpath).
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Defined.
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Global Instance singleton_fin_equiv a : IsEquiv (singleton_fin_equiv' a).
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Global Instance singleton_fin_equiv a `{IsHSet A} : IsEquiv (singleton_fin_equiv' a).
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Proof. apply _. Defined.
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Definition singleton : closedSingleton Bfin.
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Definition singleton `{IsHSet A} : closedSingleton Bfin.
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Proof.
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intros a.
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simple refine (Build_Finite _ 1 _).
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apply tr.
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apply tr.
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symmetry.
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refine (BuildEquiv _ _ (singleton_fin_equiv' a) _).
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Defined.
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@@ -48,9 +48,8 @@ Section finite_hott.
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Definition decidable_empty_finite : hasDecidableEmpty Bfin.
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Proof.
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intros X Y.
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destruct Y as [n Xn].
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destruct Y as [n f].
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strip_truncations.
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destruct Xn as [f [g fg gf adj]].
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destruct n.
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- refine (tr(inl _)).
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apply path_forall. intro z.
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@@ -59,10 +58,10 @@ Section finite_hott.
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contradiction (f (z;p)).
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* contradiction.
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- refine (tr(inr _)).
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apply (tr(g(inr tt))).
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apply (tr(f^-1(inr tt))).
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Defined.
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Lemma no_union
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Lemma no_union `{IsHSet A}
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(f : forall (X Y : Sub A),
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Bfin X -> Bfin Y -> Bfin (X ∪ Y))
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(a b : A) :
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@@ -133,121 +132,125 @@ Section finite_hott.
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** refine (px @ _ @ py^). symmetry. auto.
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** apply (px @ py^).
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Defined.
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Section empty.
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Variable (X : A -> hProp)
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(Xequiv : {a : A & a ∈ X} <~> Fin 0).
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Lemma X_empty : X = ∅.
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Proof.
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apply path_forall.
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intro z.
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apply path_iff_hprop ; try contradiction.
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intro x.
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destruct Xequiv as [f fequiv].
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contradiction (f(z;x)).
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Defined.
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End empty.
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Section split.
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Variable (X : A -> hProp)
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(n : nat)
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(Xequiv : {a : A & a ∈ X} <~> Fin n + Unit).
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Definition split : {X' : A -> hProp & {a : A & a ∈ X'} <~> Fin n}.
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Proof.
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destruct Xequiv as [f [g fg gf adj]].
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unfold Sect in *.
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pose (fun x : A => sig (fun y : Fin n => x = (g(inl y)).1 )) as X'.
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assert (forall a : A, IsHProp (X' a)).
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{
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intros.
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unfold X'.
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apply hprop_allpath.
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intros [x px] [y py].
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simple refine (path_sigma _ _ _ _ _).
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* cbn.
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pose (f(g(inl x))) as fgx.
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cut (g(inl x) = g(inl y)).
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{
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intros q.
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pose (ap f q).
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rewrite !fg in p.
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refine (path_sum_inl _ p).
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}
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apply path_sigma with (px^ @ py).
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apply path_ishprop.
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* apply path_ishprop.
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}
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pose (fun a => BuildhProp(X' a)) as Y.
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exists Y.
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unfold Y, X'.
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cbn.
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unshelve esplit.
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- intros [a [y p]]. apply y.
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- apply isequiv_biinv.
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unshelve esplit.
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* exists (fun x => (( (g(inl x)).1 ;(x;idpath)))).
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unfold Sect.
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intros [a [y p]].
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apply path_sigma with p^.
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simpl.
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rewrite transport_sigma.
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simple refine (path_sigma _ _ _ _ _) ; simpl.
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** rewrite transport_const ; reflexivity.
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** apply path_ishprop.
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* exists (fun x => (( (g(inl x)).1 ;(x;idpath)))).
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unfold Sect.
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intros x.
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reflexivity.
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Defined.
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Definition new_el : {a' : A & forall z, X z =
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lor (split.1 z) (merely (z = a'))}.
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Proof.
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exists ((Xequiv^-1 (inr tt)).1).
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intros.
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apply path_iff_hprop.
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- intros Xz.
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pose (Xequiv (z;Xz)) as fz.
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pose (c := fz).
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assert (c = fz). reflexivity.
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destruct c as [fz1 | fz2].
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* refine (tr(inl _)).
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unfold split ; simpl.
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exists fz1.
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rewrite X0.
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unfold fz.
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destruct Xequiv as [? [? ? sect ?]].
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compute.
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rewrite sect.
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reflexivity.
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* refine (tr(inr(tr _))).
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destruct fz2.
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rewrite X0.
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unfold fz.
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rewrite eissect.
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reflexivity.
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- intros X0.
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strip_truncations.
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destruct X0 as [Xl | Xr].
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* unfold split in Xl ; simpl in Xl.
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destruct Xequiv as [f [g fg gf adj]].
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destruct Xl as [m p].
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rewrite p.
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apply (g (inl m)).2.
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* strip_truncations.
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rewrite Xr.
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apply ((Xequiv^-1(inr tt)).2).
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Defined.
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End split.
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End finite_hott.
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Arguments Bfin {_} _.
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Section empty.
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Variable (A : Type).
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Variable (X : A -> hProp)
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(Xequiv : {a : A & a ∈ X} <~> Fin 0).
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Context `{Univalence}.
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Lemma X_empty : X = ∅.
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Proof.
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apply path_forall.
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intro z.
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apply path_iff_hprop ; try contradiction.
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intro x.
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destruct Xequiv as [f fequiv].
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contradiction (f(z;x)).
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Defined.
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End empty.
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Section Bfin_not_set.
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(* TODO: This should go into the HoTT library or in some other places *)
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Lemma ap_inl_path_sum_inl {A B} (x y : A) (p : inl x = inl y) :
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ap inl (path_sum_inl B p) = p.
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Proof.
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transitivity (@path_sum _ B (inl x) (inl y) (path_sum_inl B p));
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[ | apply (eisretr_path_sum _) ].
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destruct (path_sum_inl B p).
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reflexivity.
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Defined.
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Lemma ap_equiv {A B} (f : A <~> B) {x y : A} (p : x = y) :
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ap (f^-1 o f) p = eissect f x @ p @ (eissect f y)^.
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Proof. destruct p. hott_simpl. Defined.
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(* END TODO *)
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Section split.
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Context `{Univalence}.
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Variable (A : Type).
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Variable (P : A -> hProp)
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(n : nat)
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(f : {a : A & P a } <~> Fin n + Unit).
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Definition split : exists P' : Sub A, exists b : A,
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({a : A & P' a} <~> Fin n) * (forall x, P x = (P' x ∨ merely (x = b))).
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Proof.
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pose (fun x : A => sig (fun y : Fin n => x = (f^-1 (inl y)).1)) as P'.
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assert (forall x, IsHProp (P' x)).
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{
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intros a. unfold P'.
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apply hprop_allpath.
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intros [x px] [y py].
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pose (p := px^ @ py).
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assert (p2 : p # (f^-1 (inl x)).2 = (f^-1 (inl y)).2).
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{ apply path_ishprop. }
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simple refine (path_sigma' _ _ _).
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- apply path_sum_inl with Unit.
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refine (transport (fun z => z = inl y) (eisretr f (inl x)) _).
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refine (transport (fun z => _ = z) (eisretr f (inl y)) _).
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apply (ap f).
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apply path_sigma_hprop. apply p.
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- rewrite transport_paths_FlFr.
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hott_simpl; cbn.
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rewrite ap_compose.
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rewrite (ap_compose inl f^-1).
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rewrite ap_inl_path_sum_inl.
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repeat (rewrite transport_paths_FlFr; hott_simpl).
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rewrite !ap_pp.
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rewrite ap_V.
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rewrite <- !other_adj.
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rewrite <- (ap_compose f (f^-1)).
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rewrite ap_equiv.
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rewrite !ap_pp.
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rewrite ap_pr1_path_sigma_hprop.
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rewrite !concat_pp_p.
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rewrite !ap_V.
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rewrite concat_Vp.
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rewrite (concat_p_pp (ap pr1 (eissect f (f^-1 (inl x))))^).
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rewrite concat_Vp.
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hott_simpl. }
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exists (fun a => BuildhProp (P' a)).
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exists (f^-1 (inr tt)).1.
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split.
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{ unshelve eapply BuildEquiv.
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{ refine (fun x => x.2.1). }
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apply isequiv_biinv.
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unshelve esplit;
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exists (fun x => (((f^-1 (inl x)).1; (x; idpath)))).
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- intros [a [y p]]; cbn.
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eapply path_sigma with p^.
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apply path_ishprop.
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- intros x; cbn.
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reflexivity. }
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{ intros a.
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unfold P'.
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apply path_iff_hprop.
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- intros Ha.
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pose (y := f (a;Ha)).
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assert (Hy : y = f (a; Ha)) by reflexivity.
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destruct y as [y | []].
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+ refine (tr (inl _)).
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exists y.
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rewrite Hy.
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by rewrite eissect.
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+ refine (tr (inr (tr _))).
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rewrite Hy.
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by rewrite eissect.
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- intros Hstuff.
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strip_truncations.
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destruct Hstuff as [[y Hy] | Ha].
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+ rewrite Hy.
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apply (f^-1 (inl y)).2.
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+ strip_truncations.
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rewrite Ha.
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apply (f^-1 (inr tt)).2. }
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Defined.
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End split.
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Arguments Bfin {_} _.
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Arguments split {_} {_} _ _ _.
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Section Bfin_no_singletons.
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Definition S1toSig (x : S1) : {x : S1 & merely(x = base)}.
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Proof.
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exists x.
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@@ -272,9 +275,7 @@ Section Bfin_not_set.
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* apply path_ishprop.
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Defined.
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Context `{Univalence}.
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Theorem no_singleton (Hsing : Bfin {|base|}) : Empty.
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Theorem no_singleton `{Univalence} (Hsing : Bfin {|base|}) : Empty.
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Proof.
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destruct Hsing as [n equiv].
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strip_truncations.
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@@ -291,9 +292,9 @@ Section Bfin_not_set.
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apply (pos_neq_zero H').
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- apply set_path2.
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Defined.
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End Bfin_no_singletons.
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End Bfin_not_set.
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(* If A has decidable equality, then every Bfin subobject has decidable membership *)
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Section dec_membership.
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Variable (A : Type).
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Context `{DecidablePaths A} `{Univalence}.
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@@ -310,267 +311,166 @@ Section dec_membership.
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rewrite p.
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apply _.
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- intros.
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pose (new_el _ P n Hequiv) as b.
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destruct b as [b HX'].
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destruct (split _ P n Hequiv) as [X' X'equiv]. simpl in HX'.
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destruct (split P n Hequiv) as
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(P' & b & HP' & HP).
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unfold member, sub_membership.
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rewrite (HX' a).
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pose (IHn X' X'equiv) as IH.
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destruct IH as [IH | IH].
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rewrite (HP a).
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destruct (IHn P' HP') as [IH | IH].
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+ left. apply (tr (inl IH)).
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+ destruct (dec (a = b)) as [Hab | Hab].
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left. apply (tr (inr (tr Hab))).
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right. intros α. strip_truncations.
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destruct α as [β | γ]; [ | strip_truncations]; contradiction.
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destruct α as [? | ?]; [ | strip_truncations]; contradiction.
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Defined.
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End dec_membership.
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Section cowd.
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Section bfin_kfin.
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Variable (A : Type).
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Context `{Univalence}.
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Definition bfin_to_kfin : forall (P : Sub A), Bfin P -> Kf_sub _ P.
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Proof.
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intros P [n f].
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strip_truncations.
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revert f. revert P.
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induction n; intros P f.
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- exists ∅.
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apply path_forall; intro a; simpl.
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apply path_iff_hprop; [ | contradiction ].
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intros p.
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apply (f (a;p)).
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- destruct (split P n f) as
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(P' & b & HP' & HP).
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destruct (IHn P' HP') as [Y HY].
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exists (Y ∪ {|b|}).
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apply path_forall; intro a. simpl.
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rewrite <- HY.
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apply HP.
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Defined.
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End bfin_kfin.
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Section kfin_bfin.
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Variable (A : Type).
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Context `{DecidablePaths A} `{Univalence}.
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Definition cow := { X : Sub A | Bfin X}.
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Definition empty_cow : cow.
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Proof.
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exists empty. apply empty_finite.
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Defined.
|
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|
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Definition add_cow : forall a : A, cow -> cow.
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Proof.
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intros a [X PX].
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exists (fun z => lor (X z) (merely (z = a))).
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destruct (dec (a ∈ X)) as [Ha | Ha];
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destruct PX as [n PX];
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strip_truncations.
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||||
- (* a ∈ X *)
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||||
exists n. apply tr.
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||||
transitivity ({a : A & a ∈ X}); [ | apply PX ].
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||||
apply equiv_functor_sigma_id.
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||||
intro a'. eapply equiv_iff_hprop_uncurried ; split; cbn.
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||||
+ intros Ha'. strip_truncations.
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||||
destruct Ha' as [HXa' | Haa'].
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||||
* assumption.
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||||
* strip_truncations. rewrite Haa'. assumption.
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+ intros HXa'. apply tr.
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left. assumption.
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- (* a ∉ X *)
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exists (S n). apply tr.
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destruct PX as [f [g Hfg Hgf adj]].
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||||
unshelve esplit.
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||||
+ intros [a' Ha']. cbn in Ha'.
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||||
destruct (dec (a' = a)) as [Haa' | Haa'].
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* right. apply tt.
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||||
* assert (X a') as HXa'.
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||||
{ strip_truncations.
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destruct Ha' as [Ha' | Ha']; [ assumption | ].
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||||
strip_truncations. by (contradiction (Haa' Ha')). }
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||||
apply (inl (f (a';HXa'))).
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+ apply isequiv_biinv; simpl.
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||||
unshelve esplit; simpl.
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||||
* unfold Sect; simpl.
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||||
simple refine (_;_).
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||||
{ destruct 1 as [M | ?].
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||||
- destruct (g M) as [a' Ha'].
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||||
exists a'. apply tr.
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||||
by left.
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||||
- exists a. apply (tr (inr (tr idpath))). }
|
||||
simpl. intros [a' Ha'].
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||||
strip_truncations.
|
||||
destruct Ha' as [HXa' | Haa']; simpl;
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||||
destruct (dec (a' = a)); simpl.
|
||||
** apply path_sigma' with p^. apply path_ishprop.
|
||||
** rewrite Hgf; cbn. done.
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||||
** apply path_sigma' with p^. apply path_ishprop.
|
||||
** rewrite Hgf; cbn. apply path_sigma' with idpath. apply path_ishprop.
|
||||
* unfold Sect; simpl.
|
||||
simple refine (_;_).
|
||||
{ destruct 1 as [M | ?].
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||||
- destruct (g M) as [a' Ha'].
|
||||
exists a'. apply tr.
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||||
by left.
|
||||
- exists a. apply (tr (inr (tr idpath))). }
|
||||
simpl. intros [M | [] ].
|
||||
** destruct (dec (_ = a)) as [Haa' | Haa']; simpl.
|
||||
{ destruct (g M) as [a' Ha']. rewrite Haa' in Ha'. by contradiction. }
|
||||
{ f_ap. }
|
||||
** destruct (dec (a = a)); try by contradiction.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Theorem cowy
|
||||
(P : cow -> hProp)
|
||||
(doge : P empty_cow)
|
||||
(koeientaart : forall a c, P c -> P (add_cow a c))
|
||||
:
|
||||
forall X : cow, P X.
|
||||
Proof.
|
||||
intros [X [n FX]]. strip_truncations.
|
||||
revert X FX.
|
||||
induction n; intros X FX.
|
||||
- pose (HX_emp:= X_empty _ X FX).
|
||||
assert ((X; Build_Finite _ 0 (tr FX)) = empty_cow) as HX.
|
||||
{ apply path_sigma' with HX_emp. apply path_ishprop. }
|
||||
rewrite HX. assumption.
|
||||
- pose (a' := new_el _ X n FX).
|
||||
destruct a' as [a' Ha'].
|
||||
destruct (split _ X n FX) as [X' FX'].
|
||||
pose (X'cow := (X'; Build_Finite _ n (tr FX')) : cow).
|
||||
assert ((X; Build_Finite _ (n.+1) (tr FX)) = add_cow a' X'cow) as ℵ.
|
||||
{ simple refine (path_sigma' _ _ _); [ | apply path_ishprop ].
|
||||
apply path_forall. intros a.
|
||||
unfold X'cow.
|
||||
specialize (Ha' a). rewrite Ha'. simpl. reflexivity. }
|
||||
rewrite ℵ.
|
||||
apply koeientaart.
|
||||
apply IHn.
|
||||
Defined.
|
||||
|
||||
Definition bfin_to_kfin : forall (X : Sub A), Bfin X -> Kf_sub _ X.
|
||||
Proof.
|
||||
intros X BFinX.
|
||||
unfold Bfin in BFinX.
|
||||
destruct BFinX as [n iso].
|
||||
strip_truncations.
|
||||
revert iso ; revert X.
|
||||
induction n ; unfold Kf_sub, Kf_sub_intern.
|
||||
- intros.
|
||||
exists ∅.
|
||||
apply path_forall.
|
||||
intro z.
|
||||
simpl in *.
|
||||
apply path_iff_hprop ; try contradiction.
|
||||
destruct iso as [f f_equiv].
|
||||
apply (fun Xz => f(z;Xz)).
|
||||
- intros.
|
||||
simpl in *.
|
||||
destruct (new_el _ X n iso) as [a HXX'].
|
||||
destruct (split _ X n iso) as [X' X'equiv].
|
||||
destruct (IHn X' X'equiv) as [Y HY].
|
||||
exists (Y ∪ {|a|}).
|
||||
unfold map in *.
|
||||
apply path_forall.
|
||||
intro z.
|
||||
rewrite union_isIn.
|
||||
rewrite <- (ap (fun h => h z) HY).
|
||||
rewrite HXX'.
|
||||
cbn.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma kfin_is_bfin : @closedUnion A Bfin.
|
||||
Lemma bfin_union : @closedUnion A Bfin.
|
||||
Proof.
|
||||
intros X Y HX HY.
|
||||
pose (Xcow := (X; HX) : cow).
|
||||
pose (Ycow := (Y; HY) : cow).
|
||||
simple refine (cowy (fun C => Bfin (C.1 ∪ Y)) _ _ Xcow); simpl.
|
||||
- assert ((fun a => Trunc (-1) (Empty + Y a)) = (fun a => Y a)) as Help.
|
||||
{ apply path_forall. intros z; simpl.
|
||||
apply path_iff_ishprop.
|
||||
+ intros; strip_truncations; auto.
|
||||
destruct X0; auto. destruct e.
|
||||
+ intros ?. apply tr. right; assumption.
|
||||
(* TODO FIX THIS with sum_empty_l *)
|
||||
}
|
||||
rewrite Help. apply HY.
|
||||
- intros a [X' HX'] [n FX'Y]. strip_truncations.
|
||||
destruct (dec(a ∈ X')) as [HaX' | HaX'].
|
||||
* exists n. apply tr.
|
||||
transitivity ({a : A & Trunc (-1) (X' a + Y a)}); [| assumption ].
|
||||
apply equiv_functor_sigma_id. intro a'.
|
||||
apply equiv_iff_hprop.
|
||||
{ intros Q. strip_truncations.
|
||||
destruct Q as [Q | Q].
|
||||
- strip_truncations.
|
||||
apply tr. left.
|
||||
destruct Q ; auto.
|
||||
strip_truncations. rewrite t; assumption.
|
||||
- apply (tr (inr Q)). }
|
||||
{ intros Q. strip_truncations.
|
||||
destruct Q as [Q | Q]; apply tr.
|
||||
- left. apply tr. left. done.
|
||||
- right. done. }
|
||||
* destruct (dec (a ∈ Y)) as [HaY | HaY ].
|
||||
** exists n. apply tr.
|
||||
transitivity ({a : A & Trunc (-1) (X' a + Y a)}); [| assumption ].
|
||||
apply equiv_functor_sigma_id. intro a'.
|
||||
apply equiv_iff_hprop.
|
||||
{ intros Q. strip_truncations.
|
||||
destruct Q as [Q | Q].
|
||||
- strip_truncations.
|
||||
apply tr.
|
||||
destruct Q.
|
||||
left. auto.
|
||||
right. strip_truncations. rewrite t; assumption.
|
||||
- apply (tr (inr Q)). }
|
||||
{ intros Q. strip_truncations.
|
||||
destruct Q as [Q | Q]; apply tr.
|
||||
- left. apply tr. left. done.
|
||||
- right. done. }
|
||||
** exists (n.+1). apply tr.
|
||||
destruct FX'Y as [f [g Hfg Hgf adj]].
|
||||
unshelve esplit.
|
||||
{ intros [a' Ha']. cbn in Ha'.
|
||||
destruct (dec (BuildhProp (a' = a))) as [Ha'a | Ha'a].
|
||||
- right. apply tt.
|
||||
- left. refine (f (a';_)).
|
||||
destruct HX as [n fX].
|
||||
strip_truncations.
|
||||
revert fX. revert X.
|
||||
induction n; intros X fX.
|
||||
- destruct HY as [m fY]. strip_truncations.
|
||||
exists m. apply tr.
|
||||
transitivity {a : A & a ∈ Y}; [ | assumption ].
|
||||
apply equiv_functor_sigma_id.
|
||||
intros a.
|
||||
apply equiv_iff_hprop.
|
||||
* intros Ha. strip_truncations.
|
||||
destruct Ha as [Ha | Ha]; [ | apply Ha ].
|
||||
contradiction (fX (a;Ha)).
|
||||
* intros Ha. apply tr. by right.
|
||||
- destruct (split X n fX) as
|
||||
(X' & b & HX' & HX).
|
||||
assert (Bfin X') by (eexists; apply (tr HX')).
|
||||
destruct (dec (b ∈ X')) as [HX'b | HX'b].
|
||||
+ cut (X ∪ Y = X' ∪ Y).
|
||||
{ intros HXY. rewrite HXY.
|
||||
by apply IHn. }
|
||||
apply path_forall. intro a.
|
||||
unfold union, sub_union, lattice.max_fun.
|
||||
apply path_iff_hprop.
|
||||
* intros Ha.
|
||||
strip_truncations.
|
||||
destruct Ha as [HXa | HYa]; [ | apply tr; by right ].
|
||||
rewrite HX in HXa.
|
||||
strip_truncations.
|
||||
destruct HXa as [HX'a | Hab];
|
||||
[ | strip_truncations ]; apply tr; left.
|
||||
** done.
|
||||
** rewrite Hab. apply HX'b.
|
||||
* intros Ha.
|
||||
strip_truncations. apply tr.
|
||||
destruct Ha as [HXa | HYa]; [ left | by right ].
|
||||
rewrite HX. apply (tr (inl HXa)).
|
||||
+ (* b ∉ X' *)
|
||||
destruct (IHn X' HX') as [n' fw].
|
||||
strip_truncations.
|
||||
destruct (dec (b ∈ Y)) as [HYb | HYb].
|
||||
{ exists n'. apply tr.
|
||||
transitivity {a : A & a ∈ X' ∪ Y}; [ | apply fw ].
|
||||
apply equiv_functor_sigma_id. intro a.
|
||||
apply equiv_iff_hprop; cbn; simple refine (Trunc_rec _).
|
||||
{ intros [HXa | HYa].
|
||||
- rewrite HX in HXa.
|
||||
strip_truncations.
|
||||
destruct Ha' as [Ha' | Ha'].
|
||||
+ strip_truncations.
|
||||
destruct Ha' as [Ha' | Ha'].
|
||||
* apply (tr (inl Ha')).
|
||||
* strip_truncations. contradiction.
|
||||
+ apply (tr (inr Ha')). }
|
||||
{ apply isequiv_biinv; simpl.
|
||||
unshelve esplit; simpl.
|
||||
- unfold Sect; simpl.
|
||||
simple refine (_;_).
|
||||
{ destruct 1 as [M | ?].
|
||||
- destruct (g M) as [a' Ha'].
|
||||
exists a'.
|
||||
strip_truncations; apply tr.
|
||||
destruct Ha' as [Ha' | Ha'].
|
||||
+ left. apply (tr (inl Ha')).
|
||||
+ right. done.
|
||||
- exists a. apply (tr (inl (tr (inr (tr idpath))))). }
|
||||
{ intros [a' Ha']; simpl.
|
||||
strip_truncations.
|
||||
destruct Ha' as [HXa' | Haa']; simpl;
|
||||
destruct (dec (a' = a)); simpl.
|
||||
** apply path_sigma' with p^. apply path_ishprop.
|
||||
** rewrite Hgf; cbn. apply path_sigma' with idpath. apply path_ishprop.
|
||||
** apply path_sigma' with p^. apply path_ishprop.
|
||||
** rewrite Hgf; cbn. done. }
|
||||
- unfold Sect; simpl.
|
||||
simple refine (_;_).
|
||||
{ destruct 1 as [M | ?].
|
||||
- (* destruct (g M) as [a' Ha']. *)
|
||||
exists (g M).1.
|
||||
simple refine (Trunc_rec _ (g M).2).
|
||||
intros Ha'.
|
||||
apply tr.
|
||||
(* strip_truncations; apply tr. *)
|
||||
destruct Ha' as [Ha' | Ha'].
|
||||
+ left. apply (tr (inl Ha')).
|
||||
+ right. done.
|
||||
- exists a. apply (tr (inl (tr (inr (tr idpath))))). }
|
||||
simpl. intros [M | [] ].
|
||||
** destruct (dec (_ = a)) as [Haa' | Haa']; simpl.
|
||||
{ destruct (g M) as [a' Ha']. simpl in Haa'.
|
||||
strip_truncations.
|
||||
rewrite Haa' in Ha'. destruct Ha'; by contradiction. }
|
||||
{ f_ap. transitivity (f (g M)); [ | apply Hfg].
|
||||
f_ap. apply path_sigma' with idpath.
|
||||
apply path_ishprop. }
|
||||
** destruct (dec (a = a)); try by contradiction.
|
||||
reflexivity. }
|
||||
destruct HXa as [HX'a | Hab]; apply tr.
|
||||
* by left.
|
||||
* right. strip_truncations.
|
||||
rewrite Hab. apply HYb.
|
||||
- apply tr. by right. }
|
||||
{ intros [HX'a | HYa]; apply tr.
|
||||
* left. rewrite HX.
|
||||
apply (tr (inl HX'a)).
|
||||
* by right. } }
|
||||
{ exists (n'.+1).
|
||||
apply tr.
|
||||
unshelve eapply BuildEquiv.
|
||||
{ intros [a Ha]. cbn in Ha.
|
||||
destruct (dec (BuildhProp (a = b))) as [Hab | Hab].
|
||||
- right. apply tt.
|
||||
- left. refine (fw (a;_)).
|
||||
strip_truncations. apply tr.
|
||||
destruct Ha as [HXa | HYa].
|
||||
+ left. rewrite HX in HXa.
|
||||
strip_truncations.
|
||||
destruct HXa as [HX'a | Hab']; [apply HX'a |].
|
||||
strip_truncations. contradiction.
|
||||
+ right. apply HYa. }
|
||||
{ apply isequiv_biinv.
|
||||
unshelve esplit; cbn.
|
||||
- unshelve eexists.
|
||||
+ intros [m | []].
|
||||
* destruct (fw^-1 m) as [a Ha].
|
||||
exists a.
|
||||
strip_truncations. apply tr.
|
||||
destruct Ha as [HX'a | HYa]; [ left | by right ].
|
||||
rewrite HX.
|
||||
apply (tr (inl HX'a)).
|
||||
* exists b.
|
||||
rewrite HX.
|
||||
apply (tr (inl (tr (inr (tr idpath))))).
|
||||
+ intros [a Ha]; cbn.
|
||||
strip_truncations.
|
||||
simple refine (path_sigma' _ _ _); [ | apply path_ishprop ].
|
||||
destruct (H a b); cbn.
|
||||
* apply p^.
|
||||
* rewrite eissect; cbn.
|
||||
reflexivity.
|
||||
- unshelve eexists. (* TODO: Duplication!! *)
|
||||
+ intros [m | []].
|
||||
* exists (fw^-1 m).1.
|
||||
simple refine (Trunc_rec _ (fw^-1 m).2).
|
||||
intros [HX'a | HYa]; apply tr; [ left | by right ].
|
||||
rewrite HX.
|
||||
apply (tr (inl HX'a)).
|
||||
* exists b.
|
||||
rewrite HX.
|
||||
apply (tr (inl (tr (inr (tr idpath))))).
|
||||
+ intros [m | []]; cbn.
|
||||
destruct (dec (_ = b)) as [Hb | Hb]; cbn.
|
||||
{ destruct (fw^-1 m) as [a Ha]. simpl in Hb.
|
||||
simple refine (Trunc_rec _ Ha). clear Ha.
|
||||
rewrite Hb.
|
||||
intros [HX'b2 | HYb2]; contradiction. }
|
||||
{ f_ap. transitivity (fw (fw^-1 m)).
|
||||
- f_ap.
|
||||
apply path_sigma' with idpath.
|
||||
apply path_ishprop.
|
||||
- apply eisretr. }
|
||||
destruct (dec (b = b)); [ reflexivity | contradiction ]. } }
|
||||
Defined.
|
||||
|
||||
End cowd.
|
||||
|
||||
Section Kf_to_Bf.
|
||||
Context `{Univalence}.
|
||||
|
||||
Definition FSet_to_Bfin (A : Type) `{DecidablePaths A} : forall (X : FSet A), Bfin (map X).
|
||||
Definition FSet_to_Bfin : forall (X : FSet A), Bfin (map X).
|
||||
Proof.
|
||||
hinduction; try (intros; apply path_ishprop).
|
||||
- exists 0. apply tr. simpl.
|
||||
@@ -589,25 +489,25 @@ Section Kf_to_Bf.
|
||||
* exists (fun _ => (a; tr(idpath))).
|
||||
intros []. reflexivity.
|
||||
- intros Y1 Y2 HY1 HY2.
|
||||
apply kfin_is_bfin; auto.
|
||||
apply bfin_union; auto.
|
||||
Defined.
|
||||
|
||||
Instance Kf_to_Bf (X : Type) (Hfin : Kf X) `{DecidablePaths X} : Finite X.
|
||||
Proof.
|
||||
apply Kf_unfold in Hfin.
|
||||
destruct Hfin as [Y HY].
|
||||
pose (X' := FSet_to_Bfin _ Y).
|
||||
unfold Bfin in X'.
|
||||
simple refine (finite_equiv' _ _ X').
|
||||
unshelve esplit.
|
||||
- intros [a ?]. apply a.
|
||||
- apply isequiv_biinv. split.
|
||||
* exists (fun a => (a;HY a)).
|
||||
intros [b Hb].
|
||||
apply path_sigma' with idpath.
|
||||
apply path_ishprop.
|
||||
* exists (fun a => (a;HY a)).
|
||||
intros b. reflexivity.
|
||||
Defined.
|
||||
End kfin_bfin.
|
||||
|
||||
End Kf_to_Bf.
|
||||
Instance Kf_to_Bf (X : Type) `{Univalence} `{DecidablePaths X} (Hfin : Kf X) : Finite X.
|
||||
Proof.
|
||||
apply Kf_unfold in Hfin.
|
||||
destruct Hfin as [Y HY].
|
||||
pose (X' := FSet_to_Bfin _ Y).
|
||||
unfold Bfin in X'.
|
||||
simple refine (finite_equiv' _ _ X').
|
||||
unshelve esplit.
|
||||
- intros [a ?]. apply a.
|
||||
- apply isequiv_biinv. split.
|
||||
* exists (fun a => (a;HY a)).
|
||||
intros [b Hb].
|
||||
apply path_sigma' with idpath.
|
||||
apply path_ishprop.
|
||||
* exists (fun a => (a;HY a)).
|
||||
intros b. reflexivity.
|
||||
Defined.
|
||||
|
||||
@@ -63,7 +63,7 @@ Section k_fin_lemoo_projective.
|
||||
Global Instance kuratowski_projective_oo (X : Type) (Hfin : Kf X) : IsProjective X.
|
||||
Proof.
|
||||
assert (Finite X).
|
||||
{ apply Kf_to_Bf; auto.
|
||||
{ eapply Kf_to_Bf; auto.
|
||||
intros pp qq. apply LEMoo. }
|
||||
apply _.
|
||||
Defined.
|
||||
@@ -78,7 +78,7 @@ Section k_fin_lem_projective.
|
||||
Global Instance kuratowski_projective (Hfin : Kf X) : IsProjective X.
|
||||
Proof.
|
||||
assert (Finite X).
|
||||
{ apply Kf_to_Bf; auto.
|
||||
{ eapply Kf_to_Bf; auto.
|
||||
intros pp qq. apply LEM. apply _. }
|
||||
apply _.
|
||||
Defined.
|
||||
|
||||
Reference in New Issue
Block a user