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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id$ i*)
(*s Sets operations on maps *)
Require Import Bool.
Require Import Sumbool.
Require Import NArith.
Require Import Ndigits.
Require Import Ndec.
Require Import Map.
Section Dom.
Variables A B : Type.
Fixpoint MapDomRestrTo (m:Map A) : Map B -> Map A :=
match m with
| M0 => fun _:Map B => M0 A
| M1 a y =>
fun m':Map B => match MapGet B m' a with
| None => M0 A
| _ => m
end
| M2 m1 m2 =>
fun m':Map B =>
match m' with
| M0 => M0 A
| M1 a' y' =>
match MapGet A m a' with
| None => M0 A
| Some y => M1 A a' y
end
| M2 m'1 m'2 =>
makeM2 A (MapDomRestrTo m1 m'1) (MapDomRestrTo m2 m'2)
end
end.
Lemma MapDomRestrTo_semantics :
forall (m:Map A) (m':Map B),
eqm A (MapGet A (MapDomRestrTo m m'))
(fun a0:ad =>
match MapGet B m' a0 with
| None => None
| _ => MapGet A m a0
end).
Proof.
unfold eqm in |- *. simple induction m. simpl in |- *. intros. case (MapGet B m' a); trivial.
intros. simpl in |- *. elim (sumbool_of_bool (Neqb a a1)). intro H. rewrite H.
rewrite <- (Neqb_complete _ _ H). case (MapGet B m' a); try reflexivity.
intro. apply M1_semantics_1.
intro H. rewrite H. case (MapGet B m' a).
case (MapGet B m' a1); intros; exact (M1_semantics_2 A a a1 a0 H).
case (MapGet B m' a1); reflexivity.
simple induction m'. trivial.
unfold MapDomRestrTo in |- *. intros. elim (sumbool_of_bool (Neqb a a1)).
intro H1.
rewrite (Neqb_complete _ _ H1). rewrite (M1_semantics_1 B a1 a0).
case (MapGet A (M2 A m0 m1) a1); try reflexivity.
intro. apply M1_semantics_1.
intro H1. rewrite (M1_semantics_2 B a a1 a0 H1). case (MapGet A (M2 A m0 m1) a); try reflexivity.
intro. exact (M1_semantics_2 A a a1 a2 H1).
intros. change
(MapGet A (makeM2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3)) a =
match MapGet B (M2 B m2 m3) a with
| None => None
| Some _ => MapGet A (M2 A m0 m1) a
end) in |- *.
rewrite (makeM2_M2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3) a).
rewrite MapGet_M2_bit_0_if. rewrite (H0 m3 (Ndiv2 a)). rewrite (H m2 (Ndiv2 a)).
rewrite (MapGet_M2_bit_0_if B m2 m3 a). rewrite (MapGet_M2_bit_0_if A m0 m1 a).
case (Nbit0 a); reflexivity.
Qed.
Fixpoint MapDomRestrBy (m:Map A) : Map B -> Map A :=
match m with
| M0 => fun _:Map B => M0 A
| M1 a y =>
fun m':Map B => match MapGet B m' a with
| None => m
| _ => M0 A
end
| M2 m1 m2 =>
fun m':Map B =>
match m' with
| M0 => m
| M1 a' y' => MapRemove A m a'
| M2 m'1 m'2 =>
makeM2 A (MapDomRestrBy m1 m'1) (MapDomRestrBy m2 m'2)
end
end.
Lemma MapDomRestrBy_semantics :
forall (m:Map A) (m':Map B),
eqm A (MapGet A (MapDomRestrBy m m'))
(fun a0:ad =>
match MapGet B m' a0 with
| None => MapGet A m a0
| _ => None
end).
Proof.
unfold eqm in |- *. simple induction m. simpl in |- *. intros. case (MapGet B m' a); trivial.
intros. simpl in |- *. elim (sumbool_of_bool (Neqb a a1)). intro H. rewrite H.
rewrite (Neqb_complete _ _ H). case (MapGet B m' a1). trivial.
apply M1_semantics_1.
intro H. rewrite H. case (MapGet B m' a).
case (MapGet B m' a1); trivial.
rewrite (M1_semantics_2 A a a1 a0 H).
case (MapGet B m' a1); trivial.
simple induction m'. trivial.
unfold MapDomRestrBy in |- *. intros. rewrite (MapRemove_semantics A (M2 A m0 m1) a a1).
elim (sumbool_of_bool (Neqb a a1)). intro H1. rewrite H1. rewrite (Neqb_complete _ _ H1).
rewrite (M1_semantics_1 B a1 a0). reflexivity.
intro H1. rewrite H1. rewrite (M1_semantics_2 B a a1 a0 H1). reflexivity.
intros. change
(MapGet A (makeM2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3)) a =
match MapGet B (M2 B m2 m3) a with
| None => MapGet A (M2 A m0 m1) a
| Some _ => None
end) in |- *.
rewrite (makeM2_M2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3) a).
rewrite MapGet_M2_bit_0_if. rewrite (H0 m3 (Ndiv2 a)). rewrite (H m2 (Ndiv2 a)).
rewrite (MapGet_M2_bit_0_if B m2 m3 a). rewrite (MapGet_M2_bit_0_if A m0 m1 a).
case (Nbit0 a); reflexivity.
Qed.
Definition in_dom (a:ad) (m:Map A) :=
match MapGet A m a with
| None => false
| _ => true
end.
Lemma in_dom_M0 : forall a:ad, in_dom a (M0 A) = false.
Proof.
trivial.
Qed.
Lemma in_dom_M1 : forall (a a0:ad) (y:A), in_dom a0 (M1 A a y) = Neqb a a0.
Proof.
unfold in_dom in |- *. intros. simpl in |- *. case (Neqb a a0); reflexivity.
Qed.
Lemma in_dom_M1_1 : forall (a:ad) (y:A), in_dom a (M1 A a y) = true.
Proof.
intros. rewrite in_dom_M1. apply Neqb_correct.
Qed.
Lemma in_dom_M1_2 :
forall (a a0:ad) (y:A), in_dom a0 (M1 A a y) = true -> a = a0.
Proof.
intros. apply (Neqb_complete a a0). rewrite (in_dom_M1 a a0 y) in H. assumption.
Qed.
Lemma in_dom_some :
forall (m:Map A) (a:ad),
in_dom a m = true -> {y : A | MapGet A m a = Some y}.
Proof.
unfold in_dom in |- *. intros. elim (option_sum _ (MapGet A m a)). trivial.
intro H0. rewrite H0 in H. discriminate H.
Qed.
Lemma in_dom_none :
forall (m:Map A) (a:ad), in_dom a m = false -> MapGet A m a = None.
Proof.
unfold in_dom in |- *. intros. elim (option_sum _ (MapGet A m a)). intro H0. elim H0.
intros y H1. rewrite H1 in H. discriminate H.
trivial.
Qed.
Lemma in_dom_put :
forall (m:Map A) (a0:ad) (y0:A) (a:ad),
in_dom a (MapPut A m a0 y0) = orb (Neqb a a0) (in_dom a m).
Proof.
unfold in_dom in |- *. intros. rewrite (MapPut_semantics A m a0 y0 a).
elim (sumbool_of_bool (Neqb a a0)). intro H. rewrite H. rewrite (Neqb_comm a a0) in H.
rewrite H. rewrite orb_true_b. reflexivity.
intro H. rewrite H. rewrite (Neqb_comm a a0) in H. rewrite H. rewrite orb_false_b.
reflexivity.
Qed.
Lemma in_dom_put_behind :
forall (m:Map A) (a0:ad) (y0:A) (a:ad),
in_dom a (MapPut_behind A m a0 y0) = orb (Neqb a a0) (in_dom a m).
Proof.
unfold in_dom in |- *. intros. rewrite (MapPut_behind_semantics A m a0 y0 a).
elim (sumbool_of_bool (Neqb a a0)). intro H. rewrite H. rewrite (Neqb_comm a a0) in H.
rewrite H. case (MapGet A m a); reflexivity.
intro H. rewrite H. rewrite (Neqb_comm a a0) in H. rewrite H. case (MapGet A m a); trivial.
Qed.
Lemma in_dom_remove :
forall (m:Map A) (a0 a:ad),
in_dom a (MapRemove A m a0) = andb (negb (Neqb a a0)) (in_dom a m).
Proof.
unfold in_dom in |- *. intros. rewrite (MapRemove_semantics A m a0 a).
elim (sumbool_of_bool (Neqb a a0)). intro H. rewrite H. rewrite (Neqb_comm a a0) in H.
rewrite H. reflexivity.
intro H. rewrite H. rewrite (Neqb_comm a a0) in H. rewrite H.
case (MapGet A m a); reflexivity.
Qed.
Lemma in_dom_merge :
forall (m m':Map A) (a:ad),
in_dom a (MapMerge A m m') = orb (in_dom a m) (in_dom a m').
Proof.
unfold in_dom in |- *. intros. rewrite (MapMerge_semantics A m m' a).
elim (option_sum A (MapGet A m' a)). intro H. elim H. intros y H0. rewrite H0.
case (MapGet A m a); reflexivity.
intro H. rewrite H. rewrite orb_b_false. reflexivity.
Qed.
Lemma in_dom_delta :
forall (m m':Map A) (a:ad),
in_dom a (MapDelta A m m') = xorb (in_dom a m) (in_dom a m').
Proof.
unfold in_dom in |- *. intros. rewrite (MapDelta_semantics A m m' a).
elim (option_sum A (MapGet A m' a)). intro H. elim H. intros y H0. rewrite H0.
case (MapGet A m a); reflexivity.
intro H. rewrite H. case (MapGet A m a); reflexivity.
Qed.
End Dom.
Section InDom.
Variables A B : Type.
Lemma in_dom_restrto :
forall (m:Map A) (m':Map B) (a:ad),
in_dom A a (MapDomRestrTo A B m m') =
andb (in_dom A a m) (in_dom B a m').
Proof.
unfold in_dom in |- *. intros. rewrite (MapDomRestrTo_semantics A B m m' a).
elim (option_sum B (MapGet B m' a)). intro H. elim H. intros y H0. rewrite H0.
rewrite andb_b_true. reflexivity.
intro H. rewrite H. rewrite andb_b_false. reflexivity.
Qed.
Lemma in_dom_restrby :
forall (m:Map A) (m':Map B) (a:ad),
in_dom A a (MapDomRestrBy A B m m') =
andb (in_dom A a m) (negb (in_dom B a m')).
Proof.
unfold in_dom in |- *. intros. rewrite (MapDomRestrBy_semantics A B m m' a).
elim (option_sum B (MapGet B m' a)). intro H. elim H. intros y H0. rewrite H0.
unfold negb in |- *. rewrite andb_b_false. reflexivity.
intro H. rewrite H. unfold negb in |- *. rewrite andb_b_true. reflexivity.
Qed.
End InDom.
Definition FSet := Map unit.
Section FSetDefs.
Variable A : Type.
Definition in_FSet : ad -> FSet -> bool := in_dom unit.
Fixpoint MapDom (m:Map A) : FSet :=
match m with
| M0 => M0 unit
| M1 a _ => M1 unit a tt
| M2 m m' => M2 unit (MapDom m) (MapDom m')
end.
Lemma MapDom_semantics_1 :
forall (m:Map A) (a:ad) (y:A),
MapGet A m a = Some y -> in_FSet a (MapDom m) = true.
Proof.
simple induction m. intros. discriminate H.
unfold MapDom in |- *. unfold in_FSet in |- *. unfold in_dom in |- *. unfold MapGet in |- *. intros a y a0 y0.
case (Neqb a a0). trivial.
intro. discriminate H.
intros m0 H m1 H0 a y. rewrite (MapGet_M2_bit_0_if A m0 m1 a). simpl in |- *. unfold in_FSet in |- *.
unfold in_dom in |- *. rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a).
case (Nbit0 a). unfold in_FSet, in_dom in H0. intro. apply H0 with (y := y). assumption.
unfold in_FSet, in_dom in H. intro. apply H with (y := y). assumption.
Qed.
Lemma MapDom_semantics_2 :
forall (m:Map A) (a:ad),
in_FSet a (MapDom m) = true -> {y : A | MapGet A m a = Some y}.
Proof.
simple induction m. intros. discriminate H.
unfold MapDom in |- *. unfold in_FSet in |- *. unfold in_dom in |- *. unfold MapGet in |- *. intros a y a0. case (Neqb a a0).
intro. split with y. reflexivity.
intro. discriminate H.
intros m0 H m1 H0 a. rewrite (MapGet_M2_bit_0_if A m0 m1 a). simpl in |- *. unfold in_FSet in |- *.
unfold in_dom in |- *. rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a).
case (Nbit0 a). unfold in_FSet, in_dom in H0. intro. apply H0. assumption.
unfold in_FSet, in_dom in H. intro. apply H. assumption.
Qed.
Lemma MapDom_semantics_3 :
forall (m:Map A) (a:ad),
MapGet A m a = None -> in_FSet a (MapDom m) = false.
Proof.
intros. elim (sumbool_of_bool (in_FSet a (MapDom m))). intro H0.
elim (MapDom_semantics_2 m a H0). intros y H1. rewrite H in H1. discriminate H1.
trivial.
Qed.
Lemma MapDom_semantics_4 :
forall (m:Map A) (a:ad),
in_FSet a (MapDom m) = false -> MapGet A m a = None.
Proof.
intros. elim (option_sum A (MapGet A m a)). intro H0. elim H0. intros y H1.
rewrite (MapDom_semantics_1 m a y H1) in H. discriminate H.
trivial.
Qed.
Lemma MapDom_Dom :
forall (m:Map A) (a:ad), in_dom A a m = in_FSet a (MapDom m).
Proof.
intros. elim (sumbool_of_bool (in_FSet a (MapDom m))). intro H.
elim (MapDom_semantics_2 m a H). intros y H0. rewrite H. unfold in_dom in |- *. rewrite H0.
reflexivity.
intro H. rewrite H. unfold in_dom in |- *. rewrite (MapDom_semantics_4 m a H). reflexivity.
Qed.
Definition FSetUnion (s s':FSet) : FSet := MapMerge unit s s'.
Lemma in_FSet_union :
forall (s s':FSet) (a:ad),
in_FSet a (FSetUnion s s') = orb (in_FSet a s) (in_FSet a s').
Proof.
exact (in_dom_merge unit).
Qed.
Definition FSetInter (s s':FSet) : FSet := MapDomRestrTo unit unit s s'.
Lemma in_FSet_inter :
forall (s s':FSet) (a:ad),
in_FSet a (FSetInter s s') = andb (in_FSet a s) (in_FSet a s').
Proof.
exact (in_dom_restrto unit unit).
Qed.
Definition FSetDiff (s s':FSet) : FSet := MapDomRestrBy unit unit s s'.
Lemma in_FSet_diff :
forall (s s':FSet) (a:ad),
in_FSet a (FSetDiff s s') = andb (in_FSet a s) (negb (in_FSet a s')).
Proof.
exact (in_dom_restrby unit unit).
Qed.
Definition FSetDelta (s s':FSet) : FSet := MapDelta unit s s'.
Lemma in_FSet_delta :
forall (s s':FSet) (a:ad),
in_FSet a (FSetDelta s s') = xorb (in_FSet a s) (in_FSet a s').
Proof.
exact (in_dom_delta unit).
Qed.
End FSetDefs.
Lemma FSet_Dom : forall s:FSet, MapDom unit s = s.
Proof.
simple induction s. trivial.
simpl in |- *. intros a t. elim t. reflexivity.
intros. simpl in |- *. rewrite H. rewrite H0. reflexivity.
Qed.
|
