diff options
Diffstat (limited to 'theories/FSets/FMapWeakList.v')
| -rw-r--r-- | theories/FSets/FMapWeakList.v | 250 |
1 files changed, 145 insertions, 105 deletions
diff --git a/theories/FSets/FMapWeakList.v b/theories/FSets/FMapWeakList.v index ce3893e0..3a91b868 100644 --- a/theories/FSets/FMapWeakList.v +++ b/theories/FSets/FMapWeakList.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (***********************************************************************) -(* $Id: FMapWeakList.v 8639 2006-03-16 19:21:55Z letouzey $ *) +(* $Id: FMapWeakList.v 8899 2006-06-06 11:09:43Z jforest $ *) (** * Finite map library *) @@ -24,7 +24,7 @@ Arguments Scope list [type_scope]. Module Raw (X:DecidableType). -Module PX := PairDecidableType X. +Module PX := KeyDecidableType X. Import PX. Definition key := X.t. @@ -34,7 +34,7 @@ Section Elt. Variable elt : Set. -(* now in PairDecidableType: +(* now in KeyDecidableType: Definition eqk (p p':key*elt) := X.eq (fst p) (fst p'). Definition eqke (p p':key*elt) := X.eq (fst p) (fst p') /\ (snd p) = (snd p'). @@ -91,7 +91,7 @@ Qed. (** * [mem] *) -Fixpoint mem (k : key) (s : t elt) {struct s} : bool := +Function mem (k : key) (s : t elt) {struct s} : bool := match s with | nil => false | (k',_) :: l => if X.eq_dec k k' then true else mem k l @@ -100,30 +100,30 @@ Fixpoint mem (k : key) (s : t elt) {struct s} : bool := Lemma mem_1 : forall m (Hm:NoDupA m) x, In x m -> mem x m = true. Proof. intros m Hm x; generalize Hm; clear Hm. - functional induction mem x m;intros NoDup belong1;trivial. + functional induction (mem x m);intros NoDup belong1;trivial. inversion belong1. inversion H. inversion_clear NoDup. inversion_clear belong1. - inversion_clear H3. - compute in H4; destruct H4. - elim H; auto. - apply H0; auto. - exists x; auto. + inversion_clear H2. + compute in H3; destruct H3. + contradiction. + apply IHb; auto. + exists x0; auto. Qed. Lemma mem_2 : forall m (Hm:NoDupA m) x, mem x m = true -> In x m. Proof. intros m Hm x; generalize Hm; clear Hm; unfold PX.In,PX.MapsTo. - functional induction mem x m; intros NoDup hyp; try discriminate. - exists e; auto. + functional induction (mem x m); intros NoDup hyp; try discriminate. + exists _x; auto. inversion_clear NoDup. - destruct H0; auto. - exists x; auto. + destruct IHb; auto. + exists x0; auto. Qed. (** * [find] *) -Fixpoint find (k:key) (s: t elt) {struct s} : option elt := +Function find (k:key) (s: t elt) {struct s} : option elt := match s with | nil => None | (k',x)::s' => if X.eq_dec k k' then Some x else find k s' @@ -132,23 +132,23 @@ Fixpoint find (k:key) (s: t elt) {struct s} : option elt := Lemma find_2 : forall m x e, find x m = Some e -> MapsTo x e m. Proof. intros m x. unfold PX.MapsTo. - functional induction find x m;simpl;intros e' eqfind; inversion eqfind; auto. + functional induction (find x m);simpl;intros e' eqfind; inversion eqfind; auto. Qed. Lemma find_1 : forall m (Hm:NoDupA m) x e, MapsTo x e m -> find x m = Some e. Proof. intros m Hm x e; generalize Hm; clear Hm; unfold PX.MapsTo. - functional induction find x m;simpl; subst; try clear H_eq_1. + functional induction (find x m);simpl; subst; try clear H_eq_1. inversion 2. do 2 inversion_clear 1. compute in H3; destruct H3; subst; trivial. - elim H0; apply InA_eqk with (k,e); auto. + elim H; apply InA_eqk with (x,e); auto. do 2 inversion_clear 1; auto. - compute in H4; destruct H4; elim H; auto. + compute in H3; destruct H3; elim _x; auto. Qed. (* Not part of the exported specifications, used later for [combine]. *) @@ -166,7 +166,7 @@ Qed. (** * [add] *) -Fixpoint add (k : key) (x : elt) (s : t elt) {struct s} : t elt := +Function add (k : key) (x : elt) (s : t elt) {struct s} : t elt := match s with | nil => (k,x) :: nil | (k',y) :: l => if X.eq_dec k k' then (k,x)::l else (k',y)::add k x l @@ -175,26 +175,26 @@ Fixpoint add (k : key) (x : elt) (s : t elt) {struct s} : t elt := Lemma add_1 : forall m x y e, X.eq x y -> MapsTo y e (add x e m). Proof. intros m x y e; generalize y; clear y; unfold PX.MapsTo. - functional induction add x e m;simpl;auto. + functional induction (add x e m);simpl;auto. Qed. Lemma add_2 : forall m x y e e', ~ X.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m). Proof. intros m x y e e'; generalize y e; clear y e; unfold PX.MapsTo. - functional induction add x e' m;simpl;auto. - intros y' e' eqky'; inversion_clear 1. + functional induction (add x e' m);simpl;auto. + intros y' e'' eqky'; inversion_clear 1. destruct H1; simpl in *. elim eqky'; apply X.eq_trans with k'; auto. auto. - intros y' e' eqky'; inversion_clear 1; intuition. + intros y' e'' eqky'; inversion_clear 1; intuition. Qed. Lemma add_3 : forall m x y e e', ~ X.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m. Proof. intros m x y e e'. generalize y e; clear y e; unfold PX.MapsTo. - functional induction add x e' m;simpl;auto. + functional induction (add x e' m);simpl;auto. intros; apply (In_inv_3 H0); auto. constructor 2; apply (In_inv_3 H1); auto. inversion_clear 2; auto. @@ -204,12 +204,12 @@ Lemma add_3' : forall m x y e e', ~ X.eq x y -> InA eqk (y,e) (add x e' m) -> InA eqk (y,e) m. Proof. intros m x y e e'. generalize y e; clear y e. - functional induction add x e' m;simpl;auto. + functional induction (add x e' m);simpl;auto. inversion_clear 2. compute in H1; elim H; auto. inversion H1. constructor 2; inversion_clear H1; auto. - compute in H2; elim H0; auto. + compute in H2; elim H; auto. inversion_clear 2; auto. Qed. @@ -257,7 +257,7 @@ Qed. (** * [remove] *) -Fixpoint remove (k : key) (s : t elt) {struct s} : t elt := +Function remove (k : key) (s : t elt) {struct s} : t elt := match s with | nil => nil | (k',x) :: l => if X.eq_dec k k' then l else (k',x) :: remove k l @@ -266,7 +266,7 @@ Fixpoint remove (k : key) (s : t elt) {struct s} : t elt := Lemma remove_1 : forall m (Hm:NoDupA m) x y, X.eq x y -> ~ In y (remove x m). Proof. intros m Hm x y; generalize Hm; clear Hm. - functional induction remove x m;simpl;intros;auto. + functional induction (remove x m);simpl;intros;auto. red; inversion 1; inversion H1. @@ -275,14 +275,14 @@ Proof. swap H1. destruct H3 as (e,H3); unfold PX.MapsTo in H3. apply InA_eqk with (y,e); auto. - compute; apply X.eq_trans with k; auto. + compute; apply X.eq_trans with x; auto. intro H2. destruct H2 as (e,H2); inversion_clear H2. - compute in H3; destruct H3. - elim H; apply X.eq_trans with y; auto. + compute in H1; destruct H1. + elim _x; apply X.eq_trans with y; auto. inversion_clear Hm. - elim (H0 H4 H1). + elim (IHt0 H3 H). exists e; auto. Qed. @@ -290,10 +290,10 @@ Lemma remove_2 : forall m (Hm:NoDupA m) x y e, ~ X.eq x y -> MapsTo y e m -> MapsTo y e (remove x m). Proof. intros m Hm x y e; generalize Hm; clear Hm; unfold PX.MapsTo. - functional induction remove x m;auto. + functional induction (remove x m);auto. inversion_clear 3; auto. compute in H2; destruct H2. - elim H0; apply X.eq_trans with k'; auto. + elim H; apply X.eq_trans with k'; auto. inversion_clear 1; inversion_clear 2; auto. Qed. @@ -302,7 +302,7 @@ Lemma remove_3 : forall m (Hm:NoDupA m) x y e, MapsTo y e (remove x m) -> MapsTo y e m. Proof. intros m Hm x y e; generalize Hm; clear Hm; unfold PX.MapsTo. - functional induction remove x m;auto. + functional induction (remove x m);auto. do 2 inversion_clear 1; auto. Qed. @@ -310,7 +310,7 @@ Lemma remove_3' : forall m (Hm:NoDupA m) x y e, InA eqk (y,e) (remove x m) -> InA eqk (y,e) m. Proof. intros m Hm x y e; generalize Hm; clear Hm; unfold PX.MapsTo. - functional induction remove x m;auto. + functional induction (remove x m);auto. do 2 inversion_clear 1; auto. Qed. @@ -347,8 +347,7 @@ Qed. (** * [fold] *) -Fixpoint fold (A:Set)(f:key->elt->A->A)(m:t elt) {struct m} : A -> A := - fun acc => +Function fold (A:Set)(f:key->elt->A->A)(m:t elt) (acc : A) {struct m} : A := match m with | nil => acc | (k,e)::m' => fold f m' (f k e acc) @@ -357,7 +356,7 @@ Fixpoint fold (A:Set)(f:key->elt->A->A)(m:t elt) {struct m} : A -> A := Lemma fold_1 : forall m (A:Set)(i:A)(f:key->elt->A->A), fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i. Proof. - intros; functional induction fold A f m i; auto. + intros; functional induction (@fold A f m i); auto. Qed. (** * [equal] *) @@ -878,83 +877,124 @@ Module Make (X: DecidableType) <: S with Module E:=X. Module Raw := Raw X. Module E := X. - Definition key := X.t. + Definition key := E.t. Record slist (elt:Set) : Set := {this :> Raw.t elt; NoDup : NoDupA (@Raw.PX.eqk elt) this}. Definition t (elt:Set) := slist elt. - Section Elt. +Section Elt. Variable elt elt' elt'':Set. Implicit Types m : t elt. - - Definition empty := Build_slist (Raw.empty_NoDup elt). - Definition is_empty m := Raw.is_empty m.(this). - Definition add x e m := Build_slist (Raw.add_NoDup m.(NoDup) x e). - Definition find x m := Raw.find x m.(this). - Definition remove x m := Build_slist (Raw.remove_NoDup m.(NoDup) x). - Definition mem x m := Raw.mem x m.(this). + Implicit Types x y : key. + Implicit Types e : elt. + + Definition empty : t elt := Build_slist (Raw.empty_NoDup elt). + Definition is_empty m : bool := Raw.is_empty m.(this). + Definition add x e m : t elt := Build_slist (Raw.add_NoDup m.(NoDup) x e). + Definition find x m : option elt := Raw.find x m.(this). + Definition remove x m : t elt := Build_slist (Raw.remove_NoDup m.(NoDup) x). + Definition mem x m : bool := Raw.mem x m.(this). Definition map f m : t elt' := Build_slist (Raw.map_NoDup m.(NoDup) f). - Definition mapi f m : t elt' := Build_slist (Raw.mapi_NoDup m.(NoDup) f). + Definition mapi (f:key->elt->elt') m : t elt' := Build_slist (Raw.mapi_NoDup m.(NoDup) f). Definition map2 f m (m':t elt') : t elt'' := - Build_slist (Raw.map2_NoDup f m.(NoDup) m'.(NoDup)). - Definition elements m := @Raw.elements elt m.(this). - Definition fold A f m i := @Raw.fold elt A f m.(this) i. - Definition equal cmp m m' := @Raw.equal elt cmp m.(this) m'.(this). - - Definition MapsTo x e m := Raw.PX.MapsTo x e m.(this). - Definition In x m := Raw.PX.In x m.(this). - Definition Empty m := Raw.Empty m.(this). - Definition Equal cmp m m' := @Raw.Equal elt cmp m.(this) m'.(this). + Build_slist (Raw.map2_NoDup f m.(NoDup) m'.(NoDup)). + Definition elements m : list (key*elt) := @Raw.elements elt m.(this). + Definition fold (A:Set)(f:key->elt->A->A) m (i:A) : A := @Raw.fold elt A f m.(this) i. + Definition equal cmp m m' : bool := @Raw.equal elt cmp m.(this) m'.(this). + + Definition MapsTo x e m : Prop := Raw.PX.MapsTo x e m.(this). + Definition In x m : Prop := Raw.PX.In x m.(this). + Definition Empty m : Prop := Raw.Empty m.(this). + Definition Equal cmp m m' : Prop := @Raw.Equal elt cmp m.(this) m'.(this). + + Definition eq_key : (key*elt) -> (key*elt) -> Prop := @Raw.PX.eqk elt. + Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop:= @Raw.PX.eqke elt. + + Lemma MapsTo_1 : forall m x y e, E.eq x y -> MapsTo x e m -> MapsTo y e m. + Proof. intros m; exact (@Raw.PX.MapsTo_eq elt m.(this)). Qed. + + Lemma mem_1 : forall m x, In x m -> mem x m = true. + Proof. intros m; exact (@Raw.mem_1 elt m.(this) m.(NoDup)). Qed. + Lemma mem_2 : forall m x, mem x m = true -> In x m. + Proof. intros m; exact (@Raw.mem_2 elt m.(this) m.(NoDup)). Qed. + + Lemma empty_1 : Empty empty. + Proof. exact (@Raw.empty_1 elt). Qed. + + Lemma is_empty_1 : forall m, Empty m -> is_empty m = true. + Proof. intros m; exact (@Raw.is_empty_1 elt m.(this)). Qed. + Lemma is_empty_2 : forall m, is_empty m = true -> Empty m. + Proof. intros m; exact (@Raw.is_empty_2 elt m.(this)). Qed. + + Lemma add_1 : forall m x y e, E.eq x y -> MapsTo y e (add x e m). + Proof. intros m; exact (@Raw.add_1 elt m.(this)). Qed. + Lemma add_2 : forall m x y e e', ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m). + Proof. intros m; exact (@Raw.add_2 elt m.(this)). Qed. + Lemma add_3 : forall m x y e e', ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m. + Proof. intros m; exact (@Raw.add_3 elt m.(this)). Qed. + + Lemma remove_1 : forall m x y, E.eq x y -> ~ In y (remove x m). + Proof. intros m; exact (@Raw.remove_1 elt m.(this) m.(NoDup)). Qed. + Lemma remove_2 : forall m x y e, ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m). + Proof. intros m; exact (@Raw.remove_2 elt m.(this) m.(NoDup)). Qed. + Lemma remove_3 : forall m x y e, MapsTo y e (remove x m) -> MapsTo y e m. + Proof. intros m; exact (@Raw.remove_3 elt m.(this) m.(NoDup)). Qed. + + Lemma find_1 : forall m x e, MapsTo x e m -> find x m = Some e. + Proof. intros m; exact (@Raw.find_1 elt m.(this) m.(NoDup)). Qed. + Lemma find_2 : forall m x e, find x m = Some e -> MapsTo x e m. + Proof. intros m; exact (@Raw.find_2 elt m.(this)). Qed. + + Lemma elements_1 : forall m x e, MapsTo x e m -> InA eq_key_elt (x,e) (elements m). + Proof. intros m; exact (@Raw.elements_1 elt m.(this)). Qed. + Lemma elements_2 : forall m x e, InA eq_key_elt (x,e) (elements m) -> MapsTo x e m. + Proof. intros m; exact (@Raw.elements_2 elt m.(this)). Qed. + Lemma elements_3 : forall m, NoDupA eq_key (elements m). + Proof. intros m; exact (@Raw.elements_3 elt m.(this) m.(NoDup)). Qed. + + Lemma fold_1 : forall m (A : Set) (i : A) (f : key -> elt -> A -> A), + fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i. + Proof. intros m; exact (@Raw.fold_1 elt m.(this)). Qed. + + Lemma equal_1 : forall m m' cmp, Equal cmp m m' -> equal cmp m m' = true. + Proof. intros m m'; exact (@Raw.equal_1 elt m.(this) m.(NoDup) m'.(this) m'.(NoDup)). Qed. + Lemma equal_2 : forall m m' cmp, equal cmp m m' = true -> Equal cmp m m'. + Proof. intros m m'; exact (@Raw.equal_2 elt m.(this) m.(NoDup) m'.(this) m'.(NoDup)). Qed. - Definition eq_key (p p':key*elt) := X.eq (fst p) (fst p'). + End Elt. - Definition eq_key_elt (p p':key*elt) := - X.eq (fst p) (fst p') /\ (snd p) = (snd p'). - - Definition MapsTo_1 m := @Raw.PX.MapsTo_eq elt m.(this). - - Definition mem_1 m := @Raw.mem_1 elt m.(this) m.(NoDup). - Definition mem_2 m := @Raw.mem_2 elt m.(this) m.(NoDup). - - Definition empty_1 := @Raw.empty_1. - - Definition is_empty_1 m := @Raw.is_empty_1 elt m.(this). - Definition is_empty_2 m := @Raw.is_empty_2 elt m.(this). - - Definition add_1 m := @Raw.add_1 elt m.(this). - Definition add_2 m := @Raw.add_2 elt m.(this). - Definition add_3 m := @Raw.add_3 elt m.(this). + Lemma map_1 : forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:elt->elt'), + MapsTo x e m -> MapsTo x (f e) (map f m). + Proof. intros elt elt' m; exact (@Raw.map_1 elt elt' m.(this)). Qed. + Lemma map_2 : forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'), + In x (map f m) -> In x m. + Proof. intros elt elt' m; exact (@Raw.map_2 elt elt' m.(this)). Qed. + + Lemma mapi_1 : forall (elt elt':Set)(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. intros elt elt' m; exact (@Raw.mapi_1 elt elt' m.(this)). Qed. + Lemma mapi_2 : forall (elt elt':Set)(m: t elt)(x:key) + (f:key->elt->elt'), In x (mapi f m) -> In x m. + Proof. intros elt elt' m; exact (@Raw.mapi_2 elt elt' m.(this)). Qed. + + Lemma map2_1 : forall (elt elt' elt'':Set)(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'). + Proof. + intros elt elt' elt'' m m' x f; + exact (@Raw.map2_1 elt elt' elt'' f m.(this) m.(NoDup) m'.(this) m'.(NoDup) x). + Qed. + Lemma map2_2 : forall (elt elt' elt'':Set)(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. + intros elt elt' elt'' m m' x f; + exact (@Raw.map2_2 elt elt' elt'' f m.(this) m.(NoDup) m'.(this) m'.(NoDup) x). + Qed. - Definition remove_1 m := @Raw.remove_1 elt m.(this) m.(NoDup). - Definition remove_2 m := @Raw.remove_2 elt m.(this) m.(NoDup). - Definition remove_3 m := @Raw.remove_3 elt m.(this) m.(NoDup). - - Definition find_1 m := @Raw.find_1 elt m.(this) m.(NoDup). - Definition find_2 m := @Raw.find_2 elt m.(this). - - Definition elements_1 m := @Raw.elements_1 elt m.(this). - Definition elements_2 m := @Raw.elements_2 elt m.(this). - Definition elements_3 m := @Raw.elements_3 elt m.(this) m.(NoDup). - - Definition fold_1 m := @Raw.fold_1 elt m.(this). - - Definition map_1 m := @Raw.map_1 elt elt' m.(this). - Definition map_2 m := @Raw.map_2 elt elt' m.(this). - - Definition mapi_1 m := @Raw.mapi_1 elt elt' m.(this). - Definition mapi_2 m := @Raw.mapi_2 elt elt' m.(this). - - Definition map2_1 m (m':t elt') x f := - @Raw.map2_1 elt elt' elt'' f m.(this) m.(NoDup) m'.(this) m'.(NoDup) x. - Definition map2_2 m (m':t elt') x f := - @Raw.map2_2 elt elt' elt'' f m.(this) m.(NoDup) m'.(this) m'.(NoDup) x. - - Definition equal_1 m m' := @Raw.equal_1 elt m.(this) m.(NoDup) m'.(this) m'.(NoDup). - Definition equal_2 m m' := @Raw.equal_2 elt m.(this) m.(NoDup) m'.(this) m'.(NoDup). - - End Elt. End Make. - |