mirror of https://github.com/nmvdw/HITs-Examples
subset
This commit is contained in:
Leon Gondelman
parent
826b6ba233
commit
140b02e9f4
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@ -1,6 +1,5 @@
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Require Import HoTT HitTactics.
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Require Import definition.
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Section operations.
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Context {A : Type}.
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@ -48,4 +47,22 @@ intros X Y.
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apply (comprehension (fun (a : A) => isIn a X) Y).
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Defined.
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Definition subset :
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FSet A -> FSet A -> Bool.
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Proof.
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intros X Y.
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hrecursion X.
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- exact true.
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- exact (fun a => (a ∈ Y)).
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- exact andb.
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- intros. compute. destruct x; reflexivity.
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- intros x y; compute; destruct x, y; reflexivity.
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- intros x; compute; destruct x; reflexivity.
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- intros x; compute; destruct x; reflexivity.
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- intros x; cbn; destruct (x ∈ Y); reflexivity.
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Defined.
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Notation "⊆" := subset.
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End operations.
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@ -196,7 +196,8 @@ hinduction; try (intros; apply set_path2).
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Defined.
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(** assorted lattice laws *)
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Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A, intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
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Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A,
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intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
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Proof.
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hinduction; try (intros ; apply set_path2) ; cbn.
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- symmetry ; apply nl.
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@ -265,7 +266,8 @@ hinduction x; try (intros ; apply set_path2) ; cbn.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a y) ; reflexivity.
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destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a y) ;
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reflexivity.
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Defined.
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@ -345,8 +347,10 @@ hinduction X1; try (intros ; apply set_path2) ; cbn.
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cbn.
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intros Z1 Z2 P Q.
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rewrite comprehension_or.
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assert (U (U (comprehension (fun a : A => isIn a Z1) X2) (comprehension (fun a : A => isIn a Z2) X2))
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Y = U (U (comprehension (fun a : A => isIn a Z1) X2) (comprehension (fun a : A => isIn a Z2) X2))
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assert (U (U (comprehension (fun a : A => isIn a Z1) X2)
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(comprehension (fun a : A => isIn a Z2) X2))
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Y = U (U (comprehension (fun a : A => isIn a Z1) X2)
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(comprehension (fun a : A => isIn a Z2) X2))
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(U Y Y)).
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rewrite (union_idem Y).
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reflexivity.
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@ -354,8 +358,9 @@ hinduction X1; try (intros ; apply set_path2) ; cbn.
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rewrite <- assoc.
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rewrite (assoc (comprehension (fun a : A => isIn a Z2) X2)).
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rewrite Q.
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rewrite (comm (U (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)
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(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) Y).
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rewrite
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(comm (U (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)
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(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) Y).
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rewrite assoc.
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rewrite P.
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rewrite <- assoc.
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@ -364,10 +369,12 @@ hinduction X1; try (intros ; apply set_path2) ; cbn.
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rewrite <- assoc.
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rewrite assoc.
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enough (C : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) X2)
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(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)) = (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) X2)).
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(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2))
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= (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) X2)).
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rewrite C.
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enough (D : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)
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(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) = (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) Y)).
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(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y))
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= (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) Y)).
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rewrite D.
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reflexivity.
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* repeat (rewrite comprehension_or).
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@ -429,4 +436,47 @@ hrecursion X; try (intros ; apply set_path2).
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rewrite <- Q.
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Admitted.
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(* Properties about subset relation. *)
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Lemma subsect_intersection `{Funext} (X Y : FSet A) :
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subset X Y = true -> U X Y = Y.
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Proof.
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hinduction X; try (intros; apply path_forall; intro; apply set_path2).
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- intros. apply nl.
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- intros a. hinduction Y;
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try (intros; apply path_forall; intro; apply set_path2).
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(*intros. apply equiv_hprop_allpath.*)
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+ intro. cbn. contradiction (false_ne_true).
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+ intros. destruct (dec (a = a0)).
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rewrite p; apply idem.
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contradiction (false_ne_true).
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+ intros X1 X2 IH1 IH2.
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intro Ho.
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destruct (isIn a X1);
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destruct (isIn a X2).
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specialize (IH1 idpath).
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specialize (IH2 idpath).
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rewrite assoc. rewrite IH1. reflexivity.
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specialize (IH1 idpath).
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rewrite assoc. rewrite IH1. reflexivity.
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specialize (IH2 idpath).
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rewrite assoc. rewrite (comm (L a)). rewrite <- assoc. rewrite IH2.
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reflexivity.
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cbn in Ho. contradiction (false_ne_true).
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- intros X1 X2 IH1 IH2 G.
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destruct (subset X1 Y);
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destruct (subset X2 Y).
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specialize (IH1 idpath).
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specialize (IH2 idpath).
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rewrite <- assoc. rewrite IH2. rewrite IH1. reflexivity.
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specialize (IH1 idpath).
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apply IH2 in G.
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rewrite <- assoc. rewrite G. rewrite IH1. reflexivity.
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specialize (IH2 idpath).
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apply IH1 in G.
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rewrite <- assoc. rewrite IH2. rewrite G. reflexivity.
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specialize (IH1 G). specialize (IH2 G).
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rewrite <- assoc. rewrite IH2. rewrite IH1. reflexivity.
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Defined.
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End properties.
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