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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* G. Huet 1-9-95 *)
Require Import Permut Setoid.
Require Plus. (* comm. and ass. of plus *)
Set Implicit Arguments.
Section multiset_defs.
Variable A : Type.
Variable eqA : A -> A -> Prop.
Hypothesis eqA_equiv : Equivalence eqA.
Hypothesis Aeq_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Inductive multiset : Type :=
Bag : (A -> nat) -> multiset.
Definition EmptyBag := Bag (fun a:A => 0).
Definition SingletonBag (a:A) :=
Bag (fun a':A => match Aeq_dec a a' with
| left _ => 1
| right _ => 0
end).
Definition multiplicity (m:multiset) (a:A) : nat := let (f) := m in f a.
(** multiset equality *)
Definition meq (m1 m2:multiset) :=
forall a:A, multiplicity m1 a = multiplicity m2 a.
Lemma meq_refl : forall x:multiset, meq x x.
Proof.
destruct x; unfold meq; reflexivity.
Qed.
Lemma meq_trans : forall x y z:multiset, meq x y -> meq y z -> meq x z.
Proof.
unfold meq.
destruct x; destruct y; destruct z.
intros; rewrite H; auto.
Qed.
Lemma meq_sym : forall x y:multiset, meq x y -> meq y x.
Proof.
unfold meq.
destruct x; destruct y; auto.
Qed.
(** multiset union *)
Definition munion (m1 m2:multiset) :=
Bag (fun a:A => multiplicity m1 a + multiplicity m2 a).
Lemma munion_empty_left : forall x:multiset, meq x (munion EmptyBag x).
Proof.
unfold meq; unfold munion; simpl; auto.
Qed.
Lemma munion_empty_right : forall x:multiset, meq x (munion x EmptyBag).
Proof.
unfold meq; unfold munion; simpl; auto.
Qed.
Lemma munion_comm : forall x y:multiset, meq (munion x y) (munion y x).
Proof.
unfold meq; unfold multiplicity; unfold munion.
destruct x; destruct y; auto with arith.
Qed.
Lemma munion_ass :
forall x y z:multiset, meq (munion (munion x y) z) (munion x (munion y z)).
Proof.
unfold meq; unfold munion; unfold multiplicity.
destruct x; destruct y; destruct z; auto with arith.
Qed.
Lemma meq_left :
forall x y z:multiset, meq x y -> meq (munion x z) (munion y z).
Proof.
unfold meq; unfold munion; unfold multiplicity.
destruct x; destruct y; destruct z.
intros; elim H; auto with arith.
Qed.
Lemma meq_right :
forall x y z:multiset, meq x y -> meq (munion z x) (munion z y).
Proof.
unfold meq; unfold munion; unfold multiplicity.
destruct x; destruct y; destruct z.
intros; elim H; auto.
Qed.
(** Here we should make multiset an abstract datatype, by hiding [Bag],
[munion], [multiplicity]; all further properties are proved abstractly *)
Lemma munion_rotate :
forall x y z:multiset, meq (munion x (munion y z)) (munion z (munion x y)).
Proof.
intros; apply (op_rotate multiset munion meq).
apply munion_comm.
apply munion_ass.
exact meq_trans.
exact meq_sym.
trivial.
Qed.
Lemma meq_congr :
forall x y z t:multiset, meq x y -> meq z t -> meq (munion x z) (munion y t).
Proof.
intros; apply (cong_congr multiset munion meq); auto using meq_left, meq_right.
exact meq_trans.
Qed.
Lemma munion_perm_left :
forall x y z:multiset, meq (munion x (munion y z)) (munion y (munion x z)).
Proof.
intros; apply (perm_left multiset munion meq); auto using munion_comm, munion_ass, meq_left, meq_right, meq_sym.
exact meq_trans.
Qed.
Lemma multiset_twist1 :
forall x y z t:multiset,
meq (munion x (munion (munion y z) t)) (munion (munion y (munion x t)) z).
Proof.
intros; apply (twist multiset munion meq); auto using munion_comm, munion_ass, meq_sym, meq_left, meq_right.
exact meq_trans.
Qed.
Lemma multiset_twist2 :
forall x y z t:multiset,
meq (munion x (munion (munion y z) t)) (munion (munion y (munion x z)) t).
Proof.
intros; apply meq_trans with (munion (munion x (munion y z)) t).
apply meq_sym; apply munion_ass.
apply meq_left; apply munion_perm_left.
Qed.
(** specific for treesort *)
Lemma treesort_twist1 :
forall x y z t u:multiset,
meq u (munion y z) ->
meq (munion x (munion u t)) (munion (munion y (munion x t)) z).
Proof.
intros; apply meq_trans with (munion x (munion (munion y z) t)).
apply meq_right; apply meq_left; trivial.
apply multiset_twist1.
Qed.
Lemma treesort_twist2 :
forall x y z t u:multiset,
meq u (munion y z) ->
meq (munion x (munion u t)) (munion (munion y (munion x z)) t).
Proof.
intros; apply meq_trans with (munion x (munion (munion y z) t)).
apply meq_right; apply meq_left; trivial.
apply multiset_twist2.
Qed.
(** SingletonBag *)
Lemma meq_singleton : forall a a',
eqA a a' -> meq (SingletonBag a) (SingletonBag a').
Proof.
intros; red; simpl; intro a0.
destruct (Aeq_dec a a0) as [Ha|Ha]; rewrite H in Ha;
decide (Aeq_dec a' a0) with Ha; reflexivity.
Qed.
(*i theory of minter to do similarly
Require Min.
(* multiset intersection *)
Definition minter := [m1,m2:multiset]
(Bag [a:A](min (multiplicity m1 a)(multiplicity m2 a))).
i*)
End multiset_defs.
Unset Implicit Arguments.
Hint Unfold meq multiplicity: datatypes.
Hint Resolve munion_empty_right munion_comm munion_ass meq_left meq_right
munion_empty_left: datatypes.
Hint Immediate meq_sym: datatypes.
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