diff options
Diffstat (limited to 'theories/IntMap/Mapsubset.v')
| -rw-r--r-- | theories/IntMap/Mapsubset.v | 606 |
1 files changed, 606 insertions, 0 deletions
diff --git a/theories/IntMap/Mapsubset.v b/theories/IntMap/Mapsubset.v new file mode 100644 index 00000000..33b412e3 --- /dev/null +++ b/theories/IntMap/Mapsubset.v @@ -0,0 +1,606 @@ +(************************************************************************) +(* 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: Mapsubset.v,v 1.4.2.1 2004/07/16 19:31:05 herbelin Exp $ i*) + +Require Import Bool. +Require Import Sumbool. +Require Import Arith. +Require Import ZArith. +Require Import Addr. +Require Import Adist. +Require Import Addec. +Require Import Map. +Require Import Fset. +Require Import Mapaxioms. +Require Import Mapiter. + +Section MapSubsetDef. + + Variables A B : Set. + + Definition MapSubset (m:Map A) (m':Map B) := + forall a:ad, in_dom A a m = true -> in_dom B a m' = true. + + Definition MapSubset_1 (m:Map A) (m':Map B) := + match MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m with + | NONE => true + | _ => false + end. + + Definition MapSubset_2 (m:Map A) (m':Map B) := + eqmap A (MapDomRestrBy A B m m') (M0 A). + + Lemma MapSubset_imp_1 : + forall (m:Map A) (m':Map B), MapSubset m m' -> MapSubset_1 m m' = true. + Proof. + unfold MapSubset, MapSubset_1 in |- *. intros. + elim + (option_sum _ (MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m)). + intro H0. elim H0. intro r. elim r. intros a y H1. cut (negb (in_dom B a m') = true). + intro. cut (in_dom A a m = false). intro. unfold in_dom in H3. + rewrite (MapSweep_semantics_2 _ _ m a y H1) in H3. discriminate H3. + elim (sumbool_of_bool (in_dom A a m)). intro H3. rewrite (H a H3) in H2. discriminate H2. + trivial. + exact (MapSweep_semantics_1 _ _ m a y H1). + intro H0. rewrite H0. reflexivity. + Qed. + + Lemma MapSubset_1_imp : + forall (m:Map A) (m':Map B), MapSubset_1 m m' = true -> MapSubset m m'. + Proof. + unfold MapSubset, MapSubset_1 in |- *. unfold in_dom at 2 in |- *. intros. elim (option_sum _ (MapGet A m a)). + intro H1. elim H1. intros y H2. + elim + (option_sum _ (MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m)). intro H3. + elim H3. intro r. elim r. intros a' y' H4. rewrite H4 in H. discriminate H. + intro H3. cut (negb (in_dom B a m') = false). intro. rewrite (negb_intro (in_dom B a m')). + rewrite H4. reflexivity. + exact (MapSweep_semantics_3 _ _ m H3 a y H2). + intro H1. rewrite H1 in H0. discriminate H0. + Qed. + + Lemma map_dom_empty_1 : + forall m:Map A, eqmap A m (M0 A) -> forall a:ad, in_dom _ a m = false. + Proof. + unfold eqmap, eqm, in_dom in |- *. intros. rewrite (H a). reflexivity. + Qed. + + Lemma map_dom_empty_2 : + forall m:Map A, (forall a:ad, in_dom _ a m = false) -> eqmap A m (M0 A). + Proof. + unfold eqmap, eqm, in_dom in |- *. intros. + cut + (match MapGet A m a with + | NONE => false + | SOME _ => true + end = false). + case (MapGet A m a). trivial. + intros. discriminate H0. + exact (H a). + Qed. + + Lemma MapSubset_imp_2 : + forall (m:Map A) (m':Map B), MapSubset m m' -> MapSubset_2 m m'. + Proof. + unfold MapSubset, MapSubset_2 in |- *. intros. apply map_dom_empty_2. intro. rewrite in_dom_restrby. + elim (sumbool_of_bool (in_dom A a m)). intro H0. rewrite H0. rewrite (H a H0). reflexivity. + intro H0. rewrite H0. reflexivity. + Qed. + + Lemma MapSubset_2_imp : + forall (m:Map A) (m':Map B), MapSubset_2 m m' -> MapSubset m m'. + Proof. + unfold MapSubset, MapSubset_2 in |- *. intros. cut (in_dom _ a (MapDomRestrBy A B m m') = false). + rewrite in_dom_restrby. intro. elim (andb_false_elim _ _ H1). rewrite H0. + intro H2. discriminate H2. + intro H2. rewrite (negb_intro (in_dom B a m')). rewrite H2. reflexivity. + exact (map_dom_empty_1 _ H a). + Qed. + +End MapSubsetDef. + +Section MapSubsetOrder. + + Variables A B C : Set. + + Lemma MapSubset_refl : forall m:Map A, MapSubset A A m m. + Proof. + unfold MapSubset in |- *. trivial. + Qed. + + Lemma MapSubset_antisym : + forall (m:Map A) (m':Map B), + MapSubset A B m m' -> + MapSubset B A m' m -> eqmap unit (MapDom A m) (MapDom B m'). + Proof. + unfold MapSubset, eqmap, eqm in |- *. intros. elim (option_sum _ (MapGet _ (MapDom A m) a)). + intro H1. elim H1. intro t. elim t. intro H2. elim (option_sum _ (MapGet _ (MapDom B m') a)). + intro H3. elim H3. intro t'. elim t'. intro H4. rewrite H4. exact H2. + intro H3. cut (in_dom B a m' = true). intro. rewrite (MapDom_Dom B m' a) in H4. + unfold in_FSet, in_dom in H4. rewrite H3 in H4. discriminate H4. + apply H. rewrite (MapDom_Dom A m a). unfold in_FSet, in_dom in |- *. rewrite H2. reflexivity. + intro H1. elim (option_sum _ (MapGet _ (MapDom B m') a)). intro H2. elim H2. intros t H3. + cut (in_dom A a m = true). intro. rewrite (MapDom_Dom A m a) in H4. unfold in_FSet, in_dom in H4. + rewrite H1 in H4. discriminate H4. + apply H0. rewrite (MapDom_Dom B m' a). unfold in_FSet, in_dom in |- *. rewrite H3. reflexivity. + intro H2. rewrite H2. exact H1. + Qed. + + Lemma MapSubset_trans : + forall (m:Map A) (m':Map B) (m'':Map C), + MapSubset A B m m' -> MapSubset B C m' m'' -> MapSubset A C m m''. + Proof. + unfold MapSubset in |- *. intros. apply H0. apply H. assumption. + Qed. + +End MapSubsetOrder. + +Section FSubsetOrder. + + Lemma FSubset_refl : forall s:FSet, MapSubset _ _ s s. + Proof. + exact (MapSubset_refl unit). + Qed. + + Lemma FSubset_antisym : + forall s s':FSet, + MapSubset _ _ s s' -> MapSubset _ _ s' s -> eqmap unit s s'. + Proof. + intros. rewrite <- (FSet_Dom s). rewrite <- (FSet_Dom s'). + exact (MapSubset_antisym _ _ s s' H H0). + Qed. + + Lemma FSubset_trans : + forall s s' s'':FSet, + MapSubset _ _ s s' -> MapSubset _ _ s' s'' -> MapSubset _ _ s s''. + Proof. + exact (MapSubset_trans unit unit unit). + Qed. + +End FSubsetOrder. + +Section MapSubsetExtra. + + Variables A B : Set. + + Lemma MapSubset_Dom_1 : + forall (m:Map A) (m':Map B), + MapSubset A B m m' -> MapSubset unit unit (MapDom A m) (MapDom B m'). + Proof. + unfold MapSubset in |- *. intros. elim (MapDom_semantics_2 _ m a H0). intros y H1. + cut (in_dom A a m = true -> in_dom B a m' = true). intro. unfold in_dom in H2. + rewrite H1 in H2. elim (option_sum _ (MapGet B m' a)). intro H3. elim H3. + intros y' H4. exact (MapDom_semantics_1 _ m' a y' H4). + intro H3. rewrite H3 in H2. cut (false = true). intro. discriminate H4. + apply H2. reflexivity. + exact (H a). + Qed. + + Lemma MapSubset_Dom_2 : + forall (m:Map A) (m':Map B), + MapSubset unit unit (MapDom A m) (MapDom B m') -> MapSubset A B m m'. + Proof. + unfold MapSubset in |- *. intros. unfold in_dom in H0. elim (option_sum _ (MapGet A m a)). + intro H1. elim H1. intros y H2. + elim (MapDom_semantics_2 _ _ _ (H a (MapDom_semantics_1 _ _ _ _ H2))). intros y' H3. + unfold in_dom in |- *. rewrite H3. reflexivity. + intro H1. rewrite H1 in H0. discriminate H0. + Qed. + + Lemma MapSubset_1_Dom : + forall (m:Map A) (m':Map B), + MapSubset_1 A B m m' = MapSubset_1 unit unit (MapDom A m) (MapDom B m'). + Proof. + intros. elim (sumbool_of_bool (MapSubset_1 A B m m')). intro H. rewrite H. + apply sym_eq. apply MapSubset_imp_1. apply MapSubset_Dom_1. exact (MapSubset_1_imp _ _ _ _ H). + intro H. rewrite H. elim (sumbool_of_bool (MapSubset_1 unit unit (MapDom A m) (MapDom B m'))). + intro H0. + rewrite + (MapSubset_imp_1 _ _ _ _ + (MapSubset_Dom_2 _ _ (MapSubset_1_imp _ _ _ _ H0))) + in H. + discriminate H. + intro. apply sym_eq. assumption. + Qed. + + Lemma MapSubset_Put : + forall (m:Map A) (a:ad) (y:A), MapSubset A A m (MapPut A m a y). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_put. rewrite H. apply orb_b_true. + Qed. + + Lemma MapSubset_Put_mono : + forall (m:Map A) (m':Map B) (a:ad) (y:A) (y':B), + MapSubset A B m m' -> MapSubset A B (MapPut A m a y) (MapPut B m' a y'). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_put. rewrite (in_dom_put A m a y a0) in H0. + elim (orb_true_elim _ _ H0). intro H1. rewrite H1. reflexivity. + intro H1. rewrite (H _ H1). apply orb_b_true. + Qed. + + Lemma MapSubset_Put_behind : + forall (m:Map A) (a:ad) (y:A), MapSubset A A m (MapPut_behind A m a y). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_put_behind. rewrite H. apply orb_b_true. + Qed. + + Lemma MapSubset_Put_behind_mono : + forall (m:Map A) (m':Map B) (a:ad) (y:A) (y':B), + MapSubset A B m m' -> + MapSubset A B (MapPut_behind A m a y) (MapPut_behind B m' a y'). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_put_behind. + rewrite (in_dom_put_behind A m a y a0) in H0. + elim (orb_true_elim _ _ H0). intro H1. rewrite H1. reflexivity. + intro H1. rewrite (H _ H1). apply orb_b_true. + Qed. + + Lemma MapSubset_Remove : + forall (m:Map A) (a:ad), MapSubset A A (MapRemove A m a) m. + Proof. + unfold MapSubset in |- *. intros. unfold MapSubset in |- *. intros. rewrite (in_dom_remove _ m a a0) in H. + elim (andb_prop _ _ H). trivial. + Qed. + + Lemma MapSubset_Remove_mono : + forall (m:Map A) (m':Map B) (a:ad), + MapSubset A B m m' -> MapSubset A B (MapRemove A m a) (MapRemove B m' a). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_remove. rewrite (in_dom_remove A m a a0) in H0. + elim (andb_prop _ _ H0). intros. rewrite H1. rewrite (H _ H2). reflexivity. + Qed. + + Lemma MapSubset_Merge_l : + forall m m':Map A, MapSubset A A m (MapMerge A m m'). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_merge. rewrite H. reflexivity. + Qed. + + Lemma MapSubset_Merge_r : + forall m m':Map A, MapSubset A A m' (MapMerge A m m'). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_merge. rewrite H. apply orb_b_true. + Qed. + + Lemma MapSubset_Merge_mono : + forall (m m':Map A) (m'' m''':Map B), + MapSubset A B m m'' -> + MapSubset A B m' m''' -> + MapSubset A B (MapMerge A m m') (MapMerge B m'' m'''). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_merge. rewrite (in_dom_merge A m m' a) in H1. + elim (orb_true_elim _ _ H1). intro H2. rewrite (H _ H2). reflexivity. + intro H2. rewrite (H0 _ H2). apply orb_b_true. + Qed. + + Lemma MapSubset_DomRestrTo_l : + forall (m:Map A) (m':Map B), MapSubset A A (MapDomRestrTo A B m m') m. + Proof. + unfold MapSubset in |- *. intros. rewrite (in_dom_restrto _ _ m m' a) in H. elim (andb_prop _ _ H). + trivial. + Qed. + + Lemma MapSubset_DomRestrTo_r : + forall (m:Map A) (m':Map B), MapSubset A B (MapDomRestrTo A B m m') m'. + Proof. + unfold MapSubset in |- *. intros. rewrite (in_dom_restrto _ _ m m' a) in H. elim (andb_prop _ _ H). + trivial. + Qed. + + Lemma MapSubset_ext : + forall (m0 m1:Map A) (m2 m3:Map B), + eqmap A m0 m1 -> + eqmap B m2 m3 -> MapSubset A B m0 m2 -> MapSubset A B m1 m3. + Proof. + intros. apply MapSubset_2_imp. unfold MapSubset_2 in |- *. + apply eqmap_trans with (m' := MapDomRestrBy A B m0 m2). apply MapDomRestrBy_ext. apply eqmap_sym. + assumption. + apply eqmap_sym. assumption. + exact (MapSubset_imp_2 _ _ _ _ H1). + Qed. + + Variables C D : Set. + + Lemma MapSubset_DomRestrTo_mono : + forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D), + MapSubset _ _ m m'' -> + MapSubset _ _ m' m''' -> + MapSubset _ _ (MapDomRestrTo _ _ m m') (MapDomRestrTo _ _ m'' m'''). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_restrto. rewrite (in_dom_restrto A B m m' a) in H1. + elim (andb_prop _ _ H1). intros. rewrite (H _ H2). rewrite (H0 _ H3). reflexivity. + Qed. + + Lemma MapSubset_DomRestrBy_l : + forall (m:Map A) (m':Map B), MapSubset A A (MapDomRestrBy A B m m') m. + Proof. + unfold MapSubset in |- *. intros. rewrite (in_dom_restrby _ _ m m' a) in H. elim (andb_prop _ _ H). + trivial. + Qed. + + Lemma MapSubset_DomRestrBy_mono : + forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D), + MapSubset _ _ m m'' -> + MapSubset _ _ m''' m' -> + MapSubset _ _ (MapDomRestrBy _ _ m m') (MapDomRestrBy _ _ m'' m'''). + Proof. + unfold MapSubset in |- *. intros. rewrite in_dom_restrby. rewrite (in_dom_restrby A B m m' a) in H1. + elim (andb_prop _ _ H1). intros. rewrite (H _ H2). elim (sumbool_of_bool (in_dom D a m''')). + intro H4. rewrite (H0 _ H4) in H3. discriminate H3. + intro H4. rewrite H4. reflexivity. + Qed. + +End MapSubsetExtra. + +Section MapDisjointDef. + + Variables A B : Set. + + Definition MapDisjoint (m:Map A) (m':Map B) := + forall a:ad, in_dom A a m = true -> in_dom B a m' = true -> False. + + Definition MapDisjoint_1 (m:Map A) (m':Map B) := + match MapSweep A (fun (a:ad) (_:A) => in_dom B a m') m with + | NONE => true + | _ => false + end. + + Definition MapDisjoint_2 (m:Map A) (m':Map B) := + eqmap A (MapDomRestrTo A B m m') (M0 A). + + Lemma MapDisjoint_imp_1 : + forall (m:Map A) (m':Map B), MapDisjoint m m' -> MapDisjoint_1 m m' = true. + Proof. + unfold MapDisjoint, MapDisjoint_1 in |- *. intros. + elim (option_sum _ (MapSweep A (fun (a:ad) (_:A) => in_dom B a m') m)). intro H0. elim H0. + intro r. elim r. intros a y H1. cut (in_dom A a m = true -> in_dom B a m' = true -> False). + intro. unfold in_dom at 1 in H2. rewrite (MapSweep_semantics_2 _ _ _ _ _ H1) in H2. + rewrite (MapSweep_semantics_1 _ _ _ _ _ H1) in H2. elim (H2 (refl_equal _) (refl_equal _)). + exact (H a). + intro H0. rewrite H0. reflexivity. + Qed. + + Lemma MapDisjoint_1_imp : + forall (m:Map A) (m':Map B), MapDisjoint_1 m m' = true -> MapDisjoint m m'. + Proof. + unfold MapDisjoint, MapDisjoint_1 in |- *. intros. + elim (option_sum _ (MapSweep A (fun (a:ad) (_:A) => in_dom B a m') m)). intro H2. elim H2. + intro r. elim r. intros a' y' H3. rewrite H3 in H. discriminate H. + intro H2. unfold in_dom in H0. elim (option_sum _ (MapGet A m a)). intro H3. elim H3. + intros y H4. rewrite (MapSweep_semantics_3 _ _ _ H2 a y H4) in H1. discriminate H1. + intro H3. rewrite H3 in H0. discriminate H0. + Qed. + + Lemma MapDisjoint_imp_2 : + forall (m:Map A) (m':Map B), MapDisjoint m m' -> MapDisjoint_2 m m'. + Proof. + unfold MapDisjoint, MapDisjoint_2 in |- *. unfold eqmap, eqm in |- *. intros. + rewrite (MapDomRestrTo_semantics A B m m' a). + cut (in_dom A a m = true -> in_dom B a m' = true -> False). intro. + elim (option_sum _ (MapGet A m a)). intro H1. elim H1. intros y H2. unfold in_dom at 1 in H0. + elim (option_sum _ (MapGet B m' a)). intro H3. elim H3. intros y' H4. unfold in_dom at 1 in H0. + rewrite H4 in H0. rewrite H2 in H0. elim (H0 (refl_equal _) (refl_equal _)). + intro H3. rewrite H3. reflexivity. + intro H1. rewrite H1. case (MapGet B m' a); reflexivity. + exact (H a). + Qed. + + Lemma MapDisjoint_2_imp : + forall (m:Map A) (m':Map B), MapDisjoint_2 m m' -> MapDisjoint m m'. + Proof. + unfold MapDisjoint, MapDisjoint_2 in |- *. unfold eqmap, eqm in |- *. intros. elim (in_dom_some _ _ _ H0). + intros y H2. elim (in_dom_some _ _ _ H1). intros y' H3. + cut (MapGet A (MapDomRestrTo A B m m') a = NONE A). intro. + rewrite (MapDomRestrTo_semantics _ _ m m' a) in H4. rewrite H3 in H4. rewrite H2 in H4. + discriminate H4. + exact (H a). + Qed. + + Lemma Map_M0_disjoint : forall m:Map B, MapDisjoint (M0 A) m. + Proof. + unfold MapDisjoint, in_dom in |- *. intros. discriminate H. + Qed. + + Lemma Map_disjoint_M0 : forall m:Map A, MapDisjoint m (M0 B). + Proof. + unfold MapDisjoint, in_dom in |- *. intros. discriminate H0. + Qed. + +End MapDisjointDef. + +Section MapDisjointExtra. + + Variables A B : Set. + + Lemma MapDisjoint_ext : + forall (m0 m1:Map A) (m2 m3:Map B), + eqmap A m0 m1 -> + eqmap B m2 m3 -> MapDisjoint A B m0 m2 -> MapDisjoint A B m1 m3. + Proof. + intros. apply MapDisjoint_2_imp. unfold MapDisjoint_2 in |- *. + apply eqmap_trans with (m' := MapDomRestrTo A B m0 m2). apply eqmap_sym. apply MapDomRestrTo_ext. + assumption. + assumption. + exact (MapDisjoint_imp_2 _ _ _ _ H1). + Qed. + + Lemma MapMerge_disjoint : + forall m m':Map A, + MapDisjoint A A m m' -> + forall a:ad, + in_dom A a (MapMerge A m m') = + orb (andb (in_dom A a m) (negb (in_dom A a m'))) + (andb (in_dom A a m') (negb (in_dom A a m))). + Proof. + unfold MapDisjoint in |- *. intros. rewrite in_dom_merge. elim (sumbool_of_bool (in_dom A a m)). + intro H0. rewrite H0. elim (sumbool_of_bool (in_dom A a m')). intro H1. elim (H a H0 H1). + intro H1. rewrite H1. reflexivity. + intro H0. rewrite H0. simpl in |- *. rewrite andb_b_true. reflexivity. + Qed. + + Lemma MapDisjoint_M2_l : + forall (m0 m1:Map A) (m2 m3:Map B), + MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3) -> MapDisjoint A B m0 m2. + Proof. + unfold MapDisjoint, in_dom in |- *. intros. elim (option_sum _ (MapGet A m0 a)). intro H2. + elim H2. intros y H3. elim (option_sum _ (MapGet B m2 a)). intro H4. elim H4. + intros y' H5. apply (H (ad_double a)). + rewrite (MapGet_M2_bit_0_0 _ (ad_double a) (ad_double_bit_0 a) m0 m1). + rewrite (ad_double_div_2 a). rewrite H3. reflexivity. + rewrite (MapGet_M2_bit_0_0 _ (ad_double a) (ad_double_bit_0 a) m2 m3). + rewrite (ad_double_div_2 a). rewrite H5. reflexivity. + intro H4. rewrite H4 in H1. discriminate H1. + intro H2. rewrite H2 in H0. discriminate H0. + Qed. + + Lemma MapDisjoint_M2_r : + forall (m0 m1:Map A) (m2 m3:Map B), + MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3) -> MapDisjoint A B m1 m3. + Proof. + unfold MapDisjoint, in_dom in |- *. intros. elim (option_sum _ (MapGet A m1 a)). intro H2. + elim H2. intros y H3. elim (option_sum _ (MapGet B m3 a)). intro H4. elim H4. + intros y' H5. apply (H (ad_double_plus_un a)). + rewrite + (MapGet_M2_bit_0_1 _ (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) + m0 m1). + rewrite (ad_double_plus_un_div_2 a). rewrite H3. reflexivity. + rewrite + (MapGet_M2_bit_0_1 _ (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) + m2 m3). + rewrite (ad_double_plus_un_div_2 a). rewrite H5. reflexivity. + intro H4. rewrite H4 in H1. discriminate H1. + intro H2. rewrite H2 in H0. discriminate H0. + Qed. + + Lemma MapDisjoint_M2 : + forall (m0 m1:Map A) (m2 m3:Map B), + MapDisjoint A B m0 m2 -> + MapDisjoint A B m1 m3 -> MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3). + Proof. + unfold MapDisjoint, in_dom in |- *. intros. elim (sumbool_of_bool (ad_bit_0 a)). intro H3. + rewrite (MapGet_M2_bit_0_1 A a H3 m0 m1) in H1. + rewrite (MapGet_M2_bit_0_1 B a H3 m2 m3) in H2. exact (H0 (ad_div_2 a) H1 H2). + intro H3. rewrite (MapGet_M2_bit_0_0 A a H3 m0 m1) in H1. + rewrite (MapGet_M2_bit_0_0 B a H3 m2 m3) in H2. exact (H (ad_div_2 a) H1 H2). + Qed. + + Lemma MapDisjoint_M1_l : + forall (m:Map A) (a:ad) (y:B), + MapDisjoint B A (M1 B a y) m -> in_dom A a m = false. + Proof. + unfold MapDisjoint in |- *. intros. elim (sumbool_of_bool (in_dom A a m)). intro H0. + elim (H a (in_dom_M1_1 B a y) H0). + trivial. + Qed. + + Lemma MapDisjoint_M1_r : + forall (m:Map A) (a:ad) (y:B), + MapDisjoint A B m (M1 B a y) -> in_dom A a m = false. + Proof. + unfold MapDisjoint in |- *. intros. elim (sumbool_of_bool (in_dom A a m)). intro H0. + elim (H a H0 (in_dom_M1_1 B a y)). + trivial. + Qed. + + Lemma MapDisjoint_M1_conv_l : + forall (m:Map A) (a:ad) (y:B), + in_dom A a m = false -> MapDisjoint B A (M1 B a y) m. + Proof. + unfold MapDisjoint in |- *. intros. rewrite (in_dom_M1_2 B a a0 y H0) in H. rewrite H1 in H. + discriminate H. + Qed. + + Lemma MapDisjoint_M1_conv_r : + forall (m:Map A) (a:ad) (y:B), + in_dom A a m = false -> MapDisjoint A B m (M1 B a y). + Proof. + unfold MapDisjoint in |- *. intros. rewrite (in_dom_M1_2 B a a0 y H1) in H. rewrite H0 in H. + discriminate H. + Qed. + + Lemma MapDisjoint_sym : + forall (m:Map A) (m':Map B), MapDisjoint A B m m' -> MapDisjoint B A m' m. + Proof. + unfold MapDisjoint in |- *. intros. exact (H _ H1 H0). + Qed. + + Lemma MapDisjoint_empty : + forall m:Map A, MapDisjoint A A m m -> eqmap A m (M0 A). + Proof. + unfold eqmap, eqm in |- *. intros. rewrite <- (MapDomRestrTo_idempotent A m a). + exact (MapDisjoint_imp_2 A A m m H a). + Qed. + + Lemma MapDelta_disjoint : + forall m m':Map A, + MapDisjoint A A m m' -> eqmap A (MapDelta A m m') (MapMerge A m m'). + Proof. + intros. + apply eqmap_trans with + (m' := MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m')). + apply MapDelta_as_DomRestrBy. + apply eqmap_trans with (m' := MapDomRestrBy A A (MapMerge A m m') (M0 A)). + apply MapDomRestrBy_ext. apply eqmap_refl. + exact (MapDisjoint_imp_2 A A m m' H). + apply MapDomRestrBy_m_empty. + Qed. + + Variable C : Set. + + Lemma MapDomRestr_disjoint : + forall (m:Map A) (m':Map B) (m'':Map C), + MapDisjoint A B (MapDomRestrTo A C m m'') (MapDomRestrBy B C m' m''). + Proof. + unfold MapDisjoint in |- *. intros m m' m'' a. rewrite in_dom_restrto. rewrite in_dom_restrby. + intros. elim (andb_prop _ _ H). elim (andb_prop _ _ H0). intros. rewrite H4 in H2. + discriminate H2. + Qed. + + Lemma MapDelta_RestrTo_disjoint : + forall m m':Map A, + MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m m'). + Proof. + unfold MapDisjoint in |- *. intros m m' a. rewrite in_dom_delta. rewrite in_dom_restrto. + intros. elim (andb_prop _ _ H0). intros. rewrite H1 in H. rewrite H2 in H. discriminate H. + Qed. + + Lemma MapDelta_RestrTo_disjoint_2 : + forall m m':Map A, + MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m' m). + Proof. + unfold MapDisjoint in |- *. intros m m' a. rewrite in_dom_delta. rewrite in_dom_restrto. + intros. elim (andb_prop _ _ H0). intros. rewrite H1 in H. rewrite H2 in H. discriminate H. + Qed. + + Variable D : Set. + + Lemma MapSubset_Disjoint : + forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D), + MapSubset _ _ m m' -> + MapSubset _ _ m'' m''' -> + MapDisjoint _ _ m' m''' -> MapDisjoint _ _ m m''. + Proof. + unfold MapSubset, MapDisjoint in |- *. intros. exact (H1 _ (H _ H2) (H0 _ H3)). + Qed. + + Lemma MapSubset_Disjoint_l : + forall (m:Map A) (m':Map B) (m'':Map C), + MapSubset _ _ m m' -> MapDisjoint _ _ m' m'' -> MapDisjoint _ _ m m''. + Proof. + unfold MapSubset, MapDisjoint in |- *. intros. exact (H0 _ (H _ H1) H2). + Qed. + + Lemma MapSubset_Disjoint_r : + forall (m:Map A) (m'':Map C) (m''':Map D), + MapSubset _ _ m'' m''' -> + MapDisjoint _ _ m m''' -> MapDisjoint _ _ m m''. + Proof. + unfold MapSubset, MapDisjoint in |- *. intros. exact (H0 _ H1 (H _ H2)). + Qed. + +End MapDisjointExtra.
\ No newline at end of file |