2017-08-02 14:14:14 +02:00
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Require Import HoTT HitTactics.
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2017-08-03 15:07:53 +02:00
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Require Import lattice representations.definition fsets.operations extensionality Sub fsets.properties.
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2017-08-02 14:14:14 +02:00
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Section k_finite.
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Context {A : Type}.
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Context `{Univalence}.
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Definition map (X : FSet A) : Sub A := fun a => isIn a X.
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Instance map_injective : IsEmbedding map.
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Proof.
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apply isembedding_isinj_hset. (* We use the fact that both [FSet A] and [Sub A] are hSets *)
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intros X Y HXY.
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apply fset_ext.
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apply apD10. exact HXY.
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Defined.
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Definition Kf_sub_intern (B : Sub A) := exists (X : FSet A), B = map X.
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Instance Kf_sub_hprop B : IsHProp (Kf_sub_intern B).
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Proof.
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apply hprop_allpath.
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intros [X PX] [Y PY].
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assert (X = Y) as HXY.
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{ apply fset_ext. apply apD10.
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transitivity B; [ symmetry | ]; assumption. }
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apply path_sigma with HXY. simpl.
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apply set_path2.
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Defined.
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Definition Kf_sub (B : Sub A) : hProp := BuildhProp (Kf_sub_intern B).
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Definition Kf : hProp := Kf_sub (fun x => True).
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Lemma Kf_unfold : Kf <-> (exists (X : FSet A), forall (a : A), map X a).
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Proof.
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split.
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- intros [X PX]. exists X. intro a.
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rewrite <- PX. done.
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- intros [X PX]. exists X. apply path_forall; intro a.
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apply path_hprop.
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symmetry. apply if_hprop_then_equiv_Unit; [ apply _ | ].
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apply PX.
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Defined.
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End k_finite.
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2017-08-03 15:07:53 +02:00
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Section structure_k_finite.
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Context {A : Type}.
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Context `{Univalence}.
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Lemma map_union : forall X Y : FSet A, map (U X Y) = max_fun (map X) (map Y).
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Proof.
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intros.
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unfold map, max_fun.
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reflexivity.
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Defined.
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Lemma k_finite_union : @hasUnion A Kf_sub.
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Proof.
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unfold hasUnion, Kf_sub, Kf_sub_intern.
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intros.
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destruct X0 as [SX XP].
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destruct X1 as [SY YP].
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exists (U SX SY).
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rewrite map_union.
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rewrite XP, YP.
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reflexivity.
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Defined.
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Lemma k_finite_empty : @hasEmpty A Kf_sub.
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Proof.
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unfold hasEmpty, Kf_sub, Kf_sub_intern, map, empty_sub.
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exists E.
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reflexivity.
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Defined.
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Lemma k_finite_singleton : @hasSingleton A Kf_sub.
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Proof.
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unfold hasSingleton, Kf_sub, Kf_sub_intern, map, singleton.
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intro.
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exists (L a).
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cbn.
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apply path_forall.
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intro z.
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reflexivity.
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Defined.
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Lemma k_finite_hasDecidableEmpty : @hasDecidableEmpty A Kf_sub.
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Proof.
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unfold hasDecidableEmpty, hasEmpty, Kf_sub, Kf_sub_intern, map.
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intros.
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destruct X0 as [SX EX].
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rewrite EX.
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simple refine (Trunc_ind _ _ (merely_choice SX)).
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intros [SXE | H1].
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- rewrite SXE.
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apply (tr (inl idpath)).
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- apply (tr (inr H1)).
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Defined.
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End structure_k_finite.
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