diff options
Diffstat (limited to 'theories/FSets/FMapPositive.v')
| -rw-r--r-- | theories/FSets/FMapPositive.v | 102 |
1 files changed, 63 insertions, 39 deletions
diff --git a/theories/FSets/FMapPositive.v b/theories/FSets/FMapPositive.v index 44724767..9bc2a599 100644 --- a/theories/FSets/FMapPositive.v +++ b/theories/FSets/FMapPositive.v @@ -11,11 +11,12 @@ * Institution: LRI, CNRS UMR 8623 - Université Paris Sud * 91405 Orsay, France *) -(* $Id: FMapPositive.v 9862 2007-05-25 16:57:06Z letouzey $ *) +(* $Id: FMapPositive.v 10739 2008-04-01 14:45:20Z herbelin $ *) Require Import Bool. Require Import ZArith. Require Import OrderedType. +Require Import OrderedTypeEx. Require Import FMapInterface. Set Implicit Arguments. @@ -36,9 +37,12 @@ Open Local Scope positive_scope. usual order (see [OrderedTypeEx]), we use here a lexicographic order over bits, which is more natural here (lower bits are considered first). *) -Module PositiveOrderedTypeBits <: OrderedType. +Module PositiveOrderedTypeBits <: UsualOrderedType. Definition t:=positive. Definition eq:=@eq positive. + Definition eq_refl := @refl_equal t. + Definition eq_sym := @sym_eq t. + Definition eq_trans := @trans_eq t. Fixpoint bits_lt (p q:positive) { struct p } : Prop := match p, q with @@ -52,15 +56,6 @@ Module PositiveOrderedTypeBits <: OrderedType. Definition lt:=bits_lt. - Lemma eq_refl : forall x : t, eq x x. - Proof. red; auto. Qed. - - Lemma eq_sym : forall x y : t, eq x y -> eq y x. - Proof. red; auto. Qed. - - Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. - Proof. red; intros; transitivity y; auto. Qed. - Lemma bits_lt_trans : forall x y z : positive, bits_lt x y -> bits_lt y z -> bits_lt x z. Proof. induction x. @@ -171,17 +166,18 @@ Qed. Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Module E:=PositiveOrderedTypeBits. + Module ME:=KeyOrderedType E. Definition key := positive. - Inductive tree (A : Set) : Set := + Inductive tree (A : Type) := | Leaf : tree A | Node : tree A -> option A -> tree A -> tree A. Definition t := tree. Section A. - Variable A:Set. + Variable A:Type. Implicit Arguments Leaf [A]. @@ -280,6 +276,15 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Definition elements (m : t A) := xelements m xH. + (** [cardinal] *) + + Fixpoint cardinal (m : t A) : nat := + match m with + | Leaf => 0%nat + | Node l None r => (cardinal l + cardinal r)%nat + | Node l (Some _) r => S (cardinal l + cardinal r) + end. + Section CompcertSpec. Theorem gempty: @@ -560,6 +565,16 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. exact (xelements_complete i xH m v H). Qed. + Lemma cardinal_1 : + forall (m: t A), cardinal m = length (elements m). + Proof. + unfold elements. + intros m; set (p:=1); clearbody p; revert m p. + induction m; simpl; auto; intros. + rewrite (IHm1 (append p 2)), (IHm2 (append p 3)); auto. + destruct o; rewrite app_length; simpl; omega. + Qed. + End CompcertSpec. Definition MapsTo (i:positive)(v:A)(m:t A) := find i m = Some v. @@ -793,11 +808,17 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. apply xelements_sort; auto. Qed. + Lemma elements_3w : NoDupA eq_key (elements m). + Proof. + change eq_key with (@ME.eqk A). + apply ME.Sort_NoDupA; apply elements_3; auto. + Qed. + End FMapSpec. (** [map] and [mapi] *) - Variable B : Set. + Variable B : Type. Fixpoint xmapi (f : positive -> A -> B) (m : t A) (i : positive) {struct m} : t B := @@ -815,7 +836,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. End A. Lemma xgmapi: - forall (A B: Set) (f: positive -> A -> B) (i j : positive) (m: t A), + forall (A B: Type) (f: positive -> A -> B) (i j : positive) (m: t A), find i (xmapi f m j) = option_map (f (append j i)) (find i m). Proof. induction i; intros; destruct m; simpl; auto. @@ -825,7 +846,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Qed. Theorem gmapi: - forall (A B: Set) (f: positive -> A -> B) (i: positive) (m: t A), + forall (A B: Type) (f: positive -> A -> B) (i: positive) (m: t A), find i (mapi f m) = option_map (f i) (find i m). Proof. intros. @@ -836,7 +857,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Qed. Lemma mapi_1 : - forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:key->elt->elt'), + forall (elt elt':Type)(m: t elt)(x:key)(e:elt)(f:key->elt->elt'), MapsTo x e m -> exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m). Proof. @@ -851,7 +872,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Qed. Lemma mapi_2 : - forall (elt elt':Set)(m: t elt)(x:key)(f:key->elt->elt'), + forall (elt elt':Type)(m: t elt)(x:key)(f:key->elt->elt'), In x (mapi f m) -> In x m. Proof. intros. @@ -864,21 +885,21 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. simpl in *; discriminate. Qed. - Lemma map_1 : forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:elt->elt'), + Lemma map_1 : forall (elt elt':Type)(m: t elt)(x:key)(e:elt)(f:elt->elt'), MapsTo x e m -> MapsTo x (f e) (map f m). Proof. intros; unfold map. destruct (mapi_1 (fun _ => f) H); intuition. Qed. - Lemma map_2 : forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'), + Lemma map_2 : forall (elt elt':Type)(m: t elt)(x:key)(f:elt->elt'), In x (map f m) -> In x m. Proof. intros; unfold map in *; eapply mapi_2; eauto. Qed. Section map2. - Variable A B C : Set. + Variable A B C : Type. Variable f : option A -> option B -> option C. Implicit Arguments Leaf [A]. @@ -927,10 +948,10 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. End map2. - Definition map2 (elt elt' elt'':Set)(f:option elt->option elt'->option elt'') := + Definition map2 (elt elt' elt'':Type)(f:option elt->option elt'->option elt'') := _map2 (fun o1 o2 => match o1,o2 with None,None => None | _, _ => f o1 o2 end). - Lemma map2_1 : forall (elt elt' elt'':Set)(m: t elt)(m': t elt') + Lemma map2_1 : forall (elt elt' elt'':Type)(m: t elt)(m': t elt') (x:key)(f:option elt->option elt'->option elt''), In x m \/ In x m' -> find x (map2 f m m') = f (find x m) (find x m'). @@ -946,7 +967,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. destruct H; intuition; try discriminate. Qed. - Lemma map2_2 : forall (elt elt' elt'':Set)(m: t elt)(m': t elt') + Lemma map2_2 : forall (elt elt' elt'':Type)(m: t elt)(m': t elt') (x:key)(f:option elt->option elt'->option elt''), In x (map2 f m m') -> In x m \/ In x m'. Proof. @@ -962,17 +983,17 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Qed. - Definition fold (A B : Set) (f: positive -> A -> B -> B) (tr: t A) (v: B) := + Definition fold (A : Type)(B : Type) (f: positive -> A -> B -> B) (tr: t A) (v: B) := List.fold_left (fun a p => f (fst p) (snd p) a) (elements tr) v. Lemma fold_1 : - forall (A:Set)(m:t A)(B:Set)(i : B) (f : key -> A -> B -> B), + forall (A:Type)(m:t A)(B:Type)(i : B) (f : key -> A -> B -> B), fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i. Proof. intros; unfold fold; auto. Qed. - Fixpoint equal (A:Set)(cmp : A -> A -> bool)(m1 m2 : t A) {struct m1} : bool := + Fixpoint equal (A:Type)(cmp : A -> A -> bool)(m1 m2 : t A) {struct m1} : bool := match m1, m2 with | Leaf, _ => is_empty m2 | _, Leaf => is_empty m1 @@ -985,12 +1006,15 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. && equal cmp l1 l2 && equal cmp r1 r2 end. - Definition Equal (A:Set)(cmp:A->A->bool)(m m':t A) := + Definition Equal (A:Type)(m m':t A) := + forall y, find y m = find y m'. + Definition Equiv (A:Type)(eq_elt:A->A->Prop) m m' := (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true). + (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e'). + Definition Equivb (A:Type)(cmp: A->A->bool) := Equiv (Cmp cmp). - Lemma equal_1 : forall (A:Set)(m m':t A)(cmp:A->A->bool), - Equal cmp m m' -> equal cmp m m' = true. + Lemma equal_1 : forall (A:Type)(m m':t A)(cmp:A->A->bool), + Equivb cmp m m' -> equal cmp m m' = true. Proof. induction m. (* m = Leaf *) @@ -1024,11 +1048,11 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. destruct H2; red in H2; simpl in H2; discriminate. (* m' = Node *) destruct 1. - assert (Equal cmp m1 m'1). + assert (Equivb cmp m1 m'1). split. intros k; generalize (H (xO k)); unfold In, MapsTo; simpl; auto. intros k e e'; generalize (H0 (xO k) e e'); unfold In, MapsTo; simpl; auto. - assert (Equal cmp m2 m'2). + assert (Equivb cmp m2 m'2). split. intros k; generalize (H (xI k)); unfold In, MapsTo; simpl; auto. intros k e e'; generalize (H0 (xI k) e e'); unfold In, MapsTo; simpl; auto. @@ -1043,8 +1067,8 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. apply andb_true_intro; split; auto. Qed. - Lemma equal_2 : forall (A:Set)(m m':t A)(cmp:A->A->bool), - equal cmp m m' = true -> Equal cmp m m'. + Lemma equal_2 : forall (A:Type)(m m':t A)(cmp:A->A->bool), + equal cmp m m' = true -> Equivb cmp m m'. Proof. induction m. (* m = Leaf *) @@ -1103,7 +1127,7 @@ Module PositiveMapAdditionalFacts. (* Derivable from the Map interface *) Theorem gsspec: - forall (A:Set)(i j: positive) (x: A) (m: t A), + forall (A:Type)(i j: positive) (x: A) (m: t A), find i (add j x m) = if peq_dec i j then Some x else find i m. Proof. intros. @@ -1112,7 +1136,7 @@ Module PositiveMapAdditionalFacts. (* Not derivable from the Map interface *) Theorem gsident: - forall (A:Set)(i: positive) (m: t A) (v: A), + forall (A:Type)(i: positive) (m: t A) (v: A), find i m = Some v -> add i v m = m. Proof. induction i; intros; destruct m; simpl; simpl in H; try congruence. @@ -1121,7 +1145,7 @@ Module PositiveMapAdditionalFacts. Qed. Lemma xmap2_lr : - forall (A B : Set)(f g: option A -> option A -> option B)(m : t A), + forall (A B : Type)(f g: option A -> option A -> option B)(m : t A), (forall (i j : option A), f i j = g j i) -> xmap2_l f m = xmap2_r g m. Proof. @@ -1132,7 +1156,7 @@ Module PositiveMapAdditionalFacts. Qed. Theorem map2_commut: - forall (A B: Set) (f g: option A -> option A -> option B), + forall (A B: Type) (f g: option A -> option A -> option B), (forall (i j: option A), f i j = g j i) -> forall (m1 m2: t A), _map2 f m1 m2 = _map2 g m2 m1. |