listset_nodup.v 2.55 KB
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(* Copyright (c) 2012-2019, Coq-std++ developers. *)
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(* This file is distributed under the terms of the BSD license. *)
(** This file implements finite as unordered lists without duplicates.
Although this implementation is slow, it is very useful as decidable equality
is the only constraint on the carrier set. *)
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From stdpp Require Export collections list.
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Set Default Proof Using "Type".
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Record listset_nodup A := ListsetNoDup {
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  listset_nodup_car : list A; listset_nodup_prf : NoDup listset_nodup_car
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}.
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Arguments ListsetNoDup {_} _ _ : assert.
Arguments listset_nodup_car {_} _ : assert.
Arguments listset_nodup_prf {_} _ : assert.
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Section list_collection.
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Context `{EqDecision A}.
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Notation C := (listset_nodup A).

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Instance listset_nodup_elem_of: ElemOf A C := λ x l, x  listset_nodup_car l.
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Instance listset_nodup_empty: Empty C := ListsetNoDup [] (@NoDup_nil_2 _).
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Instance listset_nodup_singleton: Singleton A C := λ x,
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  ListsetNoDup [x] (NoDup_singleton x).
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Instance listset_nodup_union: Union C := λ l k,
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  let (l',Hl) := l in let (k',Hk) := k
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  in ListsetNoDup _ (NoDup_list_union _ _ Hl Hk).
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Instance listset_nodup_intersection: Intersection C := λ l k,
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  let (l',Hl) := l in let (k',Hk) := k
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  in ListsetNoDup _ (NoDup_list_intersection _ k' Hl).
Instance listset_nodup_difference: Difference C := λ l k,
  let (l',Hl) := l in let (k',Hk) := k
  in ListsetNoDup _ (NoDup_list_difference _ k' Hl).
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Instance: Collection A C.
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Proof.
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  split; [split | | ].
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  - by apply not_elem_of_nil.
  - by apply elem_of_list_singleton.
  - intros [??] [??] ?. apply elem_of_list_union.
  - intros [??] [??] ?. apply elem_of_list_intersection.
  - intros [??] [??] ?. apply elem_of_list_difference.
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Qed.

Global Instance listset_nodup_elems: Elements A C := listset_nodup_car.
Global Instance: FinCollection A C.
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Proof. split. apply _. done. by intros [??]. Qed.
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End list_collection.

Hint Extern 1 (ElemOf _ (listset_nodup _)) =>
  eapply @listset_nodup_elem_of : typeclass_instances.
Hint Extern 1 (Empty (listset_nodup _)) =>
  eapply @listset_nodup_empty : typeclass_instances.
Hint Extern 1 (Singleton _ (listset_nodup _)) =>
  eapply @listset_nodup_singleton : typeclass_instances.
Hint Extern 1 (Union (listset_nodup _)) =>
  eapply @listset_nodup_union : typeclass_instances.
Hint Extern 1 (Intersection (listset_nodup _)) =>
  eapply @listset_nodup_intersection : typeclass_instances.
Hint Extern 1 (Difference (listset_nodup _)) =>
  eapply @listset_nodup_difference : typeclass_instances.
Hint Extern 1 (Elements _ (listset_nodup _)) =>
  eapply @listset_nodup_elems : typeclass_instances.