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Diffstat (limited to 'theories/IntMap/Mapfold.v')
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diff --git a/theories/IntMap/Mapfold.v b/theories/IntMap/Mapfold.v new file mode 100644 index 00000000..641529ee --- /dev/null +++ b/theories/IntMap/Mapfold.v @@ -0,0 +1,424 @@ +(************************************************************************) +(* 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: Mapfold.v,v 1.4.2.1 2004/07/16 19:31:04 herbelin Exp $ i*) + +Require Import Bool. +Require Import Sumbool. +Require Import ZArith. +Require Import Addr. +Require Import Adist. +Require Import Addec. +Require Import Map. +Require Import Fset. +Require Import Mapaxioms. +Require Import Mapiter. +Require Import Lsort. +Require Import Mapsubset. +Require Import List. + +Section MapFoldResults. + + Variable A : Set. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Variable nleft : forall a:M, op neutral a = a. + Variable nright : forall a:M, op a neutral = a. + Variable assoc : forall a b c:M, op (op a b) c = op a (op b c). + + Lemma MapFold_ext : + forall (f:ad -> A -> M) (m m':Map A), + eqmap A m m' -> MapFold _ _ neutral op f m = MapFold _ _ neutral op f m'. + Proof. + intros. rewrite (MapFold_as_fold A M neutral op assoc nleft nright f m). + rewrite (MapFold_as_fold A M neutral op assoc nleft nright f m'). + cut (alist_of_Map A m = alist_of_Map A m'). intro. rewrite H0. reflexivity. + apply alist_canonical. unfold eqmap in H. apply eqm_trans with (f' := MapGet A m). + apply eqm_sym. apply alist_of_Map_semantics. + apply eqm_trans with (f' := MapGet A m'). assumption. + apply alist_of_Map_semantics. + apply alist_of_Map_sorts2. + apply alist_of_Map_sorts2. + Qed. + + Lemma MapFold_ext_f_1 : + forall (m:Map A) (f g:ad -> A -> M) (pf:ad -> ad), + (forall (a:ad) (y:A), MapGet _ m a = SOME _ y -> f (pf a) y = g (pf a) y) -> + MapFold1 _ _ neutral op f pf m = MapFold1 _ _ neutral op g pf m. + Proof. + simple induction m. trivial. + simpl in |- *. intros. apply H. rewrite (ad_eq_correct a). reflexivity. + intros. simpl in |- *. rewrite (H f g (fun a0:ad => pf (ad_double a0))). + rewrite (H0 f g (fun a0:ad => pf (ad_double_plus_un a0))). reflexivity. + intros. apply H1. rewrite MapGet_M2_bit_0_1. rewrite ad_double_plus_un_div_2. assumption. + apply ad_double_plus_un_bit_0. + intros. apply H1. rewrite MapGet_M2_bit_0_0. rewrite ad_double_div_2. assumption. + apply ad_double_bit_0. + Qed. + + Lemma MapFold_ext_f : + forall (f g:ad -> A -> M) (m:Map A), + (forall (a:ad) (y:A), MapGet _ m a = SOME _ y -> f a y = g a y) -> + MapFold _ _ neutral op f m = MapFold _ _ neutral op g m. + Proof. + intros. exact (MapFold_ext_f_1 m f g (fun a0:ad => a0) H). + Qed. + + Lemma MapFold1_as_Fold_1 : + forall (m:Map A) (f f':ad -> A -> M) (pf pf':ad -> ad), + (forall (a:ad) (y:A), f (pf a) y = f' (pf' a) y) -> + MapFold1 _ _ neutral op f pf m = MapFold1 _ _ neutral op f' pf' m. + Proof. + simple induction m. trivial. + intros. simpl in |- *. apply H. + intros. simpl in |- *. + rewrite + (H f f' (fun a0:ad => pf (ad_double a0)) + (fun a0:ad => pf' (ad_double a0))). + rewrite + (H0 f f' (fun a0:ad => pf (ad_double_plus_un a0)) + (fun a0:ad => pf' (ad_double_plus_un a0))). + reflexivity. + intros. apply H1. + intros. apply H1. + Qed. + + Lemma MapFold1_as_Fold : + forall (f:ad -> A -> M) (pf:ad -> ad) (m:Map A), + MapFold1 _ _ neutral op f pf m = + MapFold _ _ neutral op (fun (a:ad) (y:A) => f (pf a) y) m. + Proof. + intros. unfold MapFold in |- *. apply MapFold1_as_Fold_1. trivial. + Qed. + + Lemma MapFold1_ext : + forall (f:ad -> A -> M) (m m':Map A), + eqmap A m m' -> + forall pf:ad -> ad, + MapFold1 _ _ neutral op f pf m = MapFold1 _ _ neutral op f pf m'. + Proof. + intros. rewrite MapFold1_as_Fold. rewrite MapFold1_as_Fold. apply MapFold_ext. assumption. + Qed. + + Variable comm : forall a b:M, op a b = op b a. + + Lemma MapFold_Put_disjoint_1 : + forall (p:positive) (f:ad -> A -> M) (pf:ad -> ad) + (a1 a2:ad) (y1 y2:A), + ad_xor a1 a2 = ad_x p -> + MapFold1 A M neutral op f pf (MapPut1 A a1 y1 a2 y2 p) = + op (f (pf a1) y1) (f (pf a2) y2). + Proof. + simple induction p. intros. simpl in |- *. elim (sumbool_of_bool (ad_bit_0 a1)). intro H1. rewrite H1. + simpl in |- *. rewrite ad_div_2_double_plus_un. rewrite ad_div_2_double. apply comm. + change (ad_bit_0 a2 = negb true) in |- *. rewrite <- H1. rewrite (ad_neg_bit_0_2 _ _ _ H0). + rewrite negb_elim. reflexivity. + assumption. + intro H1. rewrite H1. simpl in |- *. rewrite ad_div_2_double. rewrite ad_div_2_double_plus_un. + reflexivity. + change (ad_bit_0 a2 = negb false) in |- *. rewrite <- H1. rewrite (ad_neg_bit_0_2 _ _ _ H0). + rewrite negb_elim. reflexivity. + assumption. + simpl in |- *. intros. elim (sumbool_of_bool (ad_bit_0 a1)). intro H1. rewrite H1. simpl in |- *. + rewrite nleft. + rewrite + (H f (fun a0:ad => pf (ad_double_plus_un a0)) ( + ad_div_2 a1) (ad_div_2 a2) y1 y2). + rewrite ad_div_2_double_plus_un. rewrite ad_div_2_double_plus_un. reflexivity. + rewrite <- (ad_same_bit_0 _ _ _ H0). assumption. + assumption. + rewrite <- ad_xor_div_2. rewrite H0. reflexivity. + intro H1. rewrite H1. simpl in |- *. rewrite nright. + rewrite + (H f (fun a0:ad => pf (ad_double a0)) (ad_div_2 a1) (ad_div_2 a2) y1 y2) + . + rewrite ad_div_2_double. rewrite ad_div_2_double. reflexivity. + rewrite <- (ad_same_bit_0 _ _ _ H0). assumption. + assumption. + rewrite <- ad_xor_div_2. rewrite H0. reflexivity. + intros. simpl in |- *. elim (sumbool_of_bool (ad_bit_0 a1)). intro H0. rewrite H0. simpl in |- *. + rewrite ad_div_2_double. rewrite ad_div_2_double_plus_un. apply comm. + assumption. + change (ad_bit_0 a2 = negb true) in |- *. rewrite <- H0. rewrite (ad_neg_bit_0_1 _ _ H). + rewrite negb_elim. reflexivity. + intro H0. rewrite H0. simpl in |- *. rewrite ad_div_2_double. rewrite ad_div_2_double_plus_un. + reflexivity. + change (ad_bit_0 a2 = negb false) in |- *. rewrite <- H0. rewrite (ad_neg_bit_0_1 _ _ H). + rewrite negb_elim. reflexivity. + assumption. + Qed. + + Lemma MapFold_Put_disjoint_2 : + forall (f:ad -> A -> M) (m:Map A) (a:ad) (y:A) (pf:ad -> ad), + MapGet A m a = NONE A -> + MapFold1 A M neutral op f pf (MapPut A m a y) = + op (f (pf a) y) (MapFold1 A M neutral op f pf m). + Proof. + simple induction m. intros. simpl in |- *. rewrite (nright (f (pf a) y)). reflexivity. + intros a1 y1 a2 y2 pf H. simpl in |- *. elim (ad_sum (ad_xor a1 a2)). intro H0. elim H0. + intros p H1. rewrite H1. rewrite comm. exact (MapFold_Put_disjoint_1 p f pf a1 a2 y1 y2 H1). + intro H0. rewrite (ad_eq_complete _ _ (ad_xor_eq_true _ _ H0)) in H. + rewrite (M1_semantics_1 A a2 y1) in H. discriminate H. + intros. elim (sumbool_of_bool (ad_bit_0 a)). intro H2. + cut (MapPut A (M2 A m0 m1) a y = M2 A m0 (MapPut A m1 (ad_div_2 a) y)). intro. + rewrite H3. simpl in |- *. rewrite (H0 (ad_div_2 a) y (fun a0:ad => pf (ad_double_plus_un a0))). + rewrite ad_div_2_double_plus_un. rewrite <- assoc. + rewrite + (comm (MapFold1 A M neutral op f (fun a0:ad => pf (ad_double a0)) m0) + (f (pf a) y)). + rewrite assoc. reflexivity. + assumption. + rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1) in H1. assumption. + simpl in |- *. elim (ad_sum a). intro H3. elim H3. intro p. elim p. intros p0 H4 H5. rewrite H5. + reflexivity. + intros p0 H4 H5. rewrite H5 in H2. discriminate H2. + intro H4. rewrite H4. reflexivity. + intro H3. rewrite H3 in H2. discriminate H2. + intro H2. cut (MapPut A (M2 A m0 m1) a y = M2 A (MapPut A m0 (ad_div_2 a) y) m1). + intro. rewrite H3. simpl in |- *. rewrite (H (ad_div_2 a) y (fun a0:ad => pf (ad_double a0))). + rewrite ad_div_2_double. rewrite <- assoc. reflexivity. + assumption. + rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1) in H1. assumption. + simpl in |- *. elim (ad_sum a). intro H3. elim H3. intro p. elim p. intros p0 H4 H5. rewrite H5 in H2. + discriminate H2. + intros p0 H4 H5. rewrite H5. reflexivity. + intro H4. rewrite H4 in H2. discriminate H2. + intro H3. rewrite H3. reflexivity. + Qed. + + Lemma MapFold_Put_disjoint : + forall (f:ad -> A -> M) (m:Map A) (a:ad) (y:A), + MapGet A m a = NONE A -> + MapFold A M neutral op f (MapPut A m a y) = + op (f a y) (MapFold A M neutral op f m). + Proof. + intros. exact (MapFold_Put_disjoint_2 f m a y (fun a0:ad => a0) H). + Qed. + + Lemma MapFold_Put_behind_disjoint_2 : + forall (f:ad -> A -> M) (m:Map A) (a:ad) (y:A) (pf:ad -> ad), + MapGet A m a = NONE A -> + MapFold1 A M neutral op f pf (MapPut_behind A m a y) = + op (f (pf a) y) (MapFold1 A M neutral op f pf m). + Proof. + intros. cut (eqmap A (MapPut_behind A m a y) (MapPut A m a y)). intro. + rewrite (MapFold1_ext f _ _ H0 pf). apply MapFold_Put_disjoint_2. assumption. + apply eqmap_trans with (m' := MapMerge A (M1 A a y) m). apply MapPut_behind_as_Merge. + apply eqmap_trans with (m' := MapMerge A m (M1 A a y)). + apply eqmap_trans with (m' := MapDelta A (M1 A a y) m). apply eqmap_sym. apply MapDelta_disjoint. + unfold MapDisjoint in |- *. unfold in_dom in |- *. simpl in |- *. intros. elim (sumbool_of_bool (ad_eq a a0)). + intro H2. rewrite (ad_eq_complete _ _ H2) in H. rewrite H in H1. discriminate H1. + intro H2. rewrite H2 in H0. discriminate H0. + apply eqmap_trans with (m' := MapDelta A m (M1 A a y)). apply MapDelta_sym. + apply MapDelta_disjoint. unfold MapDisjoint in |- *. unfold in_dom in |- *. simpl in |- *. intros. + elim (sumbool_of_bool (ad_eq a a0)). intro H2. rewrite (ad_eq_complete _ _ H2) in H. + rewrite H in H0. discriminate H0. + intro H2. rewrite H2 in H1. discriminate H1. + apply eqmap_sym. apply MapPut_as_Merge. + Qed. + + Lemma MapFold_Put_behind_disjoint : + forall (f:ad -> A -> M) (m:Map A) (a:ad) (y:A), + MapGet A m a = NONE A -> + MapFold A M neutral op f (MapPut_behind A m a y) = + op (f a y) (MapFold A M neutral op f m). + Proof. + intros. exact (MapFold_Put_behind_disjoint_2 f m a y (fun a0:ad => a0) H). + Qed. + + Lemma MapFold_Merge_disjoint_1 : + forall (f:ad -> A -> M) (m1 m2:Map A) (pf:ad -> ad), + MapDisjoint A A m1 m2 -> + MapFold1 A M neutral op f pf (MapMerge A m1 m2) = + op (MapFold1 A M neutral op f pf m1) (MapFold1 A M neutral op f pf m2). + Proof. + simple induction m1. simpl in |- *. intros. rewrite nleft. reflexivity. + intros. unfold MapMerge in |- *. apply (MapFold_Put_behind_disjoint_2 f m2 a a0 pf). + apply in_dom_none. exact (MapDisjoint_M1_l _ _ m2 a a0 H). + simple induction m2. intros. simpl in |- *. rewrite nright. reflexivity. + intros. unfold MapMerge in |- *. rewrite (MapFold_Put_disjoint_2 f (M2 A m m0) a a0 pf). apply comm. + apply in_dom_none. exact (MapDisjoint_M1_r _ _ (M2 A m m0) a a0 H1). + intros. simpl in |- *. rewrite (H m3 (fun a0:ad => pf (ad_double a0))). + rewrite (H0 m4 (fun a0:ad => pf (ad_double_plus_un a0))). + cut (forall a b c d:M, op (op a b) (op c d) = op (op a c) (op b d)). intro. apply H4. + intros. rewrite assoc. rewrite <- (assoc b c d). rewrite (comm b c). rewrite (assoc c b d). + rewrite assoc. reflexivity. + exact (MapDisjoint_M2_r _ _ _ _ _ _ H3). + exact (MapDisjoint_M2_l _ _ _ _ _ _ H3). + Qed. + + Lemma MapFold_Merge_disjoint : + forall (f:ad -> A -> M) (m1 m2:Map A), + MapDisjoint A A m1 m2 -> + MapFold A M neutral op f (MapMerge A m1 m2) = + op (MapFold A M neutral op f m1) (MapFold A M neutral op f m2). + Proof. + intros. exact (MapFold_Merge_disjoint_1 f m1 m2 (fun a0:ad => a0) H). + Qed. + +End MapFoldResults. + +Section MapFoldDistr. + + Variable A : Set. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Variable M' : Set. + Variable neutral' : M'. + Variable op' : M' -> M' -> M'. + + Variable N : Set. + + Variable times : M -> N -> M'. + + Variable absorb : forall c:N, times neutral c = neutral'. + Variable + distr : + forall (a b:M) (c:N), times (op a b) c = op' (times a c) (times b c). + + Lemma MapFold_distr_r_1 : + forall (f:ad -> A -> M) (m:Map A) (c:N) (pf:ad -> ad), + times (MapFold1 A M neutral op f pf m) c = + MapFold1 A M' neutral' op' (fun (a:ad) (y:A) => times (f a y) c) pf m. + Proof. + simple induction m. intros. exact (absorb c). + trivial. + intros. simpl in |- *. rewrite distr. rewrite H. rewrite H0. reflexivity. + Qed. + + Lemma MapFold_distr_r : + forall (f:ad -> A -> M) (m:Map A) (c:N), + times (MapFold A M neutral op f m) c = + MapFold A M' neutral' op' (fun (a:ad) (y:A) => times (f a y) c) m. + Proof. + intros. exact (MapFold_distr_r_1 f m c (fun a:ad => a)). + Qed. + +End MapFoldDistr. + +Section MapFoldDistrL. + + Variable A : Set. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Variable M' : Set. + Variable neutral' : M'. + Variable op' : M' -> M' -> M'. + + Variable N : Set. + + Variable times : N -> M -> M'. + + Variable absorb : forall c:N, times c neutral = neutral'. + Variable + distr : + forall (a b:M) (c:N), times c (op a b) = op' (times c a) (times c b). + + Lemma MapFold_distr_l : + forall (f:ad -> A -> M) (m:Map A) (c:N), + times c (MapFold A M neutral op f m) = + MapFold A M' neutral' op' (fun (a:ad) (y:A) => times c (f a y)) m. + Proof. + intros. apply MapFold_distr_r with (times := fun (a:M) (b:N) => times b a); + assumption. + Qed. + +End MapFoldDistrL. + +Section MapFoldExists. + + Variable A : Set. + + Lemma MapFold_orb_1 : + forall (f:ad -> A -> bool) (m:Map A) (pf:ad -> ad), + MapFold1 A bool false orb f pf m = + match MapSweep1 A f pf m with + | SOME _ => true + | _ => false + end. + Proof. + simple induction m. trivial. + intros a y pf. simpl in |- *. unfold MapSweep2 in |- *. case (f (pf a) y); reflexivity. + intros. simpl in |- *. rewrite (H (fun a0:ad => pf (ad_double a0))). + rewrite (H0 (fun a0:ad => pf (ad_double_plus_un a0))). + case (MapSweep1 A f (fun a0:ad => pf (ad_double a0)) m0); reflexivity. + Qed. + + Lemma MapFold_orb : + forall (f:ad -> A -> bool) (m:Map A), + MapFold A bool false orb f m = + match MapSweep A f m with + | SOME _ => true + | _ => false + end. + Proof. + intros. exact (MapFold_orb_1 f m (fun a:ad => a)). + Qed. + +End MapFoldExists. + +Section DMergeDef. + + Variable A : Set. + + Definition DMerge := + MapFold (Map A) (Map A) (M0 A) (MapMerge A) (fun (_:ad) (m:Map A) => m). + + Lemma in_dom_DMerge_1 : + forall (m:Map (Map A)) (a:ad), + in_dom A a (DMerge m) = + match MapSweep _ (fun (_:ad) (m0:Map A) => in_dom A a m0) m with + | SOME _ => true + | _ => false + end. + Proof. + unfold DMerge in |- *. intros. + rewrite + (MapFold_distr_l (Map A) (Map A) (M0 A) (MapMerge A) bool false orb ad + (in_dom A) (fun c:ad => refl_equal _) (in_dom_merge A)) + . + apply MapFold_orb. + Qed. + + Lemma in_dom_DMerge_2 : + forall (m:Map (Map A)) (a:ad), + in_dom A a (DMerge m) = true -> + {b : ad & + {m0 : Map A | MapGet _ m b = SOME _ m0 /\ in_dom A a m0 = true}}. + Proof. + intros m a. rewrite in_dom_DMerge_1. + elim + (option_sum _ + (MapSweep (Map A) (fun (_:ad) (m0:Map A) => in_dom A a m0) m)). + intro H. elim H. intro r. elim r. intros b m0 H0. intro. split with b. split with m0. + split. exact (MapSweep_semantics_2 _ _ _ _ _ H0). + exact (MapSweep_semantics_1 _ _ _ _ _ H0). + intro H. rewrite H. intro. discriminate H0. + Qed. + + Lemma in_dom_DMerge_3 : + forall (m:Map (Map A)) (a b:ad) (m0:Map A), + MapGet _ m a = SOME _ m0 -> + in_dom A b m0 = true -> in_dom A b (DMerge m) = true. + Proof. + intros m a b m0 H H0. rewrite in_dom_DMerge_1. + elim + (MapSweep_semantics_4 _ (fun (_:ad) (m'0:Map A) => in_dom A b m'0) _ _ _ + H H0). + intros a' H1. elim H1. intros m'0 H2. rewrite H2. reflexivity. + Qed. + +End DMergeDef.
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