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Diffstat (limited to 'theories/IntMap/Fset.v')
| -rw-r--r-- | theories/IntMap/Fset.v | 371 |
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diff --git a/theories/IntMap/Fset.v b/theories/IntMap/Fset.v new file mode 100644 index 00000000..8d217be9 --- /dev/null +++ b/theories/IntMap/Fset.v @@ -0,0 +1,371 @@ +(************************************************************************) +(* 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: Fset.v,v 1.5.2.1 2004/07/16 19:31:04 herbelin Exp $ i*) + +(*s Sets operations on maps *) + +Require Import Bool. +Require Import Sumbool. +Require Import ZArith. +Require Import Addr. +Require Import Adist. +Require Import Addec. +Require Import Map. + +Section Dom. + + Variables A B : Set. + + 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 A + | _ => 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 (ad_eq a a1)). intro H. rewrite H. + rewrite <- (ad_eq_complete _ _ H). case (MapGet B m' a). reflexivity. + intro. apply M1_semantics_1. + intro H. rewrite H. case (MapGet B m' a). + case (MapGet B m' a1); reflexivity. + case (MapGet B m' a1); intros; exact (M1_semantics_2 A a a1 a0 H). + simple induction m'. trivial. + unfold MapDomRestrTo in |- *. intros. elim (sumbool_of_bool (ad_eq a a1)). + intro H1. + rewrite (ad_eq_complete _ _ H1). rewrite (M1_semantics_1 B a1 a0). + case (MapGet A (M2 A m0 m1) a1). 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). 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 A + | 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 (ad_div_2 a)). rewrite (H m2 (ad_div_2 a)). + rewrite (MapGet_M2_bit_0_if B m2 m3 a). rewrite (MapGet_M2_bit_0_if A m0 m1 a). + case (ad_bit_0 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 A + end). + Proof. + unfold eqm in |- *. simple induction m. simpl in |- *. intros. case (MapGet B m' a); trivial. + intros. simpl in |- *. elim (sumbool_of_bool (ad_eq a a1)). intro H. rewrite H. + rewrite (ad_eq_complete _ _ H). case (MapGet B m' a1). apply M1_semantics_1. + trivial. + intro H. rewrite H. case (MapGet B m' a). rewrite (M1_semantics_2 A a a1 a0 H). + case (MapGet B m' a1); trivial. + 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 (ad_eq a a1)). intro H1. rewrite H1. rewrite (ad_eq_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 A + end) in |- *. + rewrite (makeM2_M2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3) a). + rewrite MapGet_M2_bit_0_if. rewrite (H0 m3 (ad_div_2 a)). rewrite (H m2 (ad_div_2 a)). + rewrite (MapGet_M2_bit_0_if B m2 m3 a). rewrite (MapGet_M2_bit_0_if A m0 m1 a). + case (ad_bit_0 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) = ad_eq a a0. + Proof. + unfold in_dom in |- *. intros. simpl in |- *. case (ad_eq 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 ad_eq_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 (ad_eq_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 A 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 A. + 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 (ad_eq a a0) (in_dom a m). + Proof. + unfold in_dom in |- *. intros. rewrite (MapPut_semantics A m a0 y0 a). + elim (sumbool_of_bool (ad_eq a a0)). intro H. rewrite H. rewrite (ad_eq_comm a a0) in H. + rewrite H. rewrite orb_true_b. reflexivity. + intro H. rewrite H. rewrite (ad_eq_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 (ad_eq 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 (ad_eq a a0)). intro H. rewrite H. rewrite (ad_eq_comm a a0) in H. + rewrite H. case (MapGet A m a); reflexivity. + intro H. rewrite H. rewrite (ad_eq_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 (ad_eq a a0)) (in_dom a m). + Proof. + unfold in_dom in |- *. intros. rewrite (MapRemove_semantics A m a0 a). + elim (sumbool_of_bool (ad_eq a a0)). intro H. rewrite H. rewrite (ad_eq_comm a a0) in H. + rewrite H. reflexivity. + intro H. rewrite H. rewrite (ad_eq_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 : Set. + + 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 : Set. + + 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 A 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 (ad_eq 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 (ad_bit_0 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 A 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 (ad_eq 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 (ad_bit_0 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 A -> 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 A. + 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.
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