aboutsummaryrefslogtreecommitdiffhomepage
diff options
context:
space:
mode:
authorGravatar notin <notin@85f007b7-540e-0410-9357-904b9bb8a0f7>2006-03-15 16:19:53 +0000
committerGravatar notin <notin@85f007b7-540e-0410-9357-904b9bb8a0f7>2006-03-15 16:19:53 +0000
commitbb3aa221cec09dc997b0f7dac7bc4e1a1b51c744 (patch)
treec650e4f52de0d687b412b4f251d85484e90372b0
parent1b299d804e74bee348b1de51f7946af67956fbb5 (diff)
Réparation de FSet (back to 8628)
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@8631 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat
-rw-r--r--theories/FSets/DecidableType.v62
-rw-r--r--theories/FSets/FMap.v2
-rw-r--r--theories/FSets/FMapInterface.v162
-rw-r--r--theories/FSets/FMapList.v482
-rw-r--r--theories/FSets/FMapWeak.v2
-rw-r--r--theories/FSets/FMapWeakInterface.v128
-rw-r--r--theories/FSets/FMapWeakList.v388
-rw-r--r--theories/FSets/FSet.v3
-rw-r--r--theories/FSets/FSetBridge.v430
-rw-r--r--theories/FSets/FSetEqProperties.v302
-rw-r--r--theories/FSets/FSetFacts.v122
-rw-r--r--theories/FSets/FSetInterface.v352
-rw-r--r--theories/FSets/FSetList.v650
-rw-r--r--theories/FSets/FSetProperties.v396
-rw-r--r--theories/FSets/FSetWeak.v2
-rw-r--r--theories/FSets/FSetWeakInterface.v164
-rw-r--r--theories/FSets/FSetWeakList.v492
-rw-r--r--theories/FSets/OrderedType.v278
18 files changed, 2208 insertions, 2209 deletions
diff --git a/theories/FSets/DecidableType.v b/theories/FSets/DecidableType.v
index 4ba2b191b..56bcb680d 100644
--- a/theories/FSets/DecidableType.v
+++ b/theories/FSets/DecidableType.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: DecidableType.v,v 1.3 2006/03/03 18:48:37 letouzey Exp $ *)
Require Export SetoidList.
Set Implicit Arguments.
@@ -16,15 +16,15 @@ Unset Strict Implicit.
Module Type DecidableType.
- Parameter t Set.
+ Parameter t : Set.
- Parameter eq t -> t -> Prop.
+ Parameter eq : t -> t -> Prop.
- Axiom eq_refl forall x : t, eq x x.
- Axiom eq_sym forall x y : t, eq x y -> eq y x.
- Axiom eq_trans forall x y z : t, eq x y -> eq y z -> eq x z.
+ Axiom eq_refl : forall x : t, eq x x.
+ Axiom eq_sym : forall x y : t, eq x y -> eq y x.
+ Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
- Parameter eq_dec forall x y : t, { eq x y } + { ~ eq x y }.
+ Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
Hint Immediate eq_sym.
Hint Resolve eq_refl eq_trans.
@@ -32,15 +32,15 @@ Module Type DecidableType.
End DecidableType.
-Module PairDecidableType(DDecidableType).
+Module PairDecidableType(D:DecidableType).
Import D.
Section Elt.
- Variable elt Set.
- Notation key=t.
+ Variable elt : Set.
+ Notation key:=t.
- Definition eqk (p p'key*elt) := eq (fst p) (fst p').
- Definition eqke (p p'key*elt) :=
+ Definition eqk (p p':key*elt) := eq (fst p) (fst p').
+ Definition eqke (p p':key*elt) :=
eq (fst p) (fst p') /\ (snd p) = (snd p').
Hint Unfold eqk eqke.
@@ -48,29 +48,29 @@ Module PairDecidableType(DDecidableType).
(* eqke is stricter than eqk *)
- Lemma eqke_eqk forall x x', eqke x x' -> eqk x x'.
+ Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.
Proof.
unfold eqk, eqke; intuition.
Qed.
(* eqk, eqke are equalities *)
- Lemma eqk_refl forall e, eqk e e.
+ Lemma eqk_refl : forall e, eqk e e.
Proof. auto. Qed.
- Lemma eqke_refl forall e, eqke e e.
+ Lemma eqke_refl : forall e, eqke e e.
Proof. auto. Qed.
- Lemma eqk_sym forall e e', eqk e e' -> eqk e' e.
+ Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e.
Proof. auto. Qed.
- Lemma eqke_sym forall e e', eqke e e' -> eqke e' e.
+ Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e.
Proof. unfold eqke; intuition. Qed.
- Lemma eqk_trans forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.
+ Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.
Proof. eauto. Qed.
- Lemma eqke_trans forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.
+ Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.
Proof.
unfold eqke; intuition; [ eauto | congruence ].
Qed.
@@ -78,26 +78,26 @@ Module PairDecidableType(DDecidableType).
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
Hint Immediate eqk_sym eqke_sym.
- Lemma InA_eqke_eqk
+ Lemma InA_eqke_eqk :
forall x m, InA eqke x m -> InA eqk x m.
Proof.
unfold eqke; induction 1; intuition.
Qed.
Hint Resolve InA_eqke_eqk.
- Lemma InA_eqk forall p q m, eqk p q -> InA eqk p m -> InA eqk q m.
+ Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m.
Proof.
intros; apply InA_eqA with p; auto; apply eqk_trans; auto.
Qed.
- Definition MapsTo (kkey)(e:elt):= InA eqke (k,e).
- Definition In k m = exists e:elt, MapsTo k e m.
+ Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
+ Definition In k m := exists e:elt, MapsTo k e m.
Hint Unfold MapsTo In.
(* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *)
- Lemma In_alt forall k l, In k l <-> exists e, InA eqk (k,e) l.
+ Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l.
Proof.
firstorder.
exists x; auto.
@@ -108,31 +108,31 @@ Module PairDecidableType(DDecidableType).
exists e; auto.
Qed.
- Lemma MapsTo_eq forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
+ Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
Proof.
intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto.
Qed.
- Lemma In_eq forall l x y, eq x y -> In x l -> In y l.
+ Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.
Proof.
destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto.
Qed.
- Lemma In_inv forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
+ Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
Proof.
inversion 1.
inversion_clear H0; eauto.
destruct H1; simpl in *; intuition.
Qed.
- Lemma In_inv_2 forall k k' e e' l,
- InA eqk (k, e) ((k', e') : l) -> ~ eq k k' -> InA eqk (k, e) l.
+ Lemma In_inv_2 : forall k k' e e' l,
+ InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l.
Proof.
inversion_clear 1; compute in H0; intuition.
Qed.
- Lemma In_inv_3 forall x x' l,
- InA eqke x (x' : l) -> ~ eqk x x' -> InA eqke x l.
+ Lemma In_inv_3 : forall x x' l,
+ InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
Proof.
inversion_clear 1; compute in H0; intuition.
Qed.
diff --git a/theories/FSets/FMap.v b/theories/FSets/FMap.v
index dac1b4396..695003968 100644
--- a/theories/FSets/FMap.v
+++ b/theories/FSets/FMap.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSet.v,v 1.2 2004/12/08 19:19:24 letouzey Exp $ *)
Require Export FMapInterface.
Require Export FMapList.
diff --git a/theories/FSets/FMapInterface.v b/theories/FSets/FMapInterface.v
index 2ab0a0d8d..828f5dc05 100644
--- a/theories/FSets/FMapInterface.v
+++ b/theories/FSets/FMapInterface.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FMapInterface.v,v 1.13 2006/02/27 15:39:43 letouzey Exp $ *)
(** * Finite map library *)
@@ -17,7 +17,7 @@ Unset Strict Implicit.
Require Import FSetInterface.
-(** When compared with Ocaml Map, this signature has been split in two
+(** When compared with Ocaml Map, this signature has been split in two:
- The first part [S] contains the usual operators (add, find, ...)
It only requires a ordered key type, the data type can be arbitrary.
The only function that asks more is [equal], whose first argument should
@@ -29,81 +29,81 @@ Require Import FSetInterface.
Module Type S.
- Declare Module E OrderedType.
+ Declare Module E : OrderedType.
- Definition key = E.t.
+ Definition key := E.t.
- Parameter t Set -> Set. (** the abstract type of maps *)
+ Parameter t : Set -> Set. (** the abstract type of maps *)
Section Types.
- Variable eltSet.
+ Variable elt:Set.
- Parameter empty t elt.
+ Parameter empty : t elt.
(** The empty map. *)
- Parameter is_empty t elt -> bool.
+ Parameter is_empty : t elt -> bool.
(** Test whether a map is empty or not. *)
- Parameter add key -> elt -> t elt -> t elt.
+ Parameter add : key -> elt -> t elt -> t elt.
(** [add x y m] returns a map containing the same bindings as [m],
plus a binding of [x] to [y]. If [x] was already bound in [m],
its previous binding disappears. *)
- Parameter find key -> t elt -> option elt.
+ Parameter find : key -> t elt -> option elt.
(** [find x m] returns the current binding of [x] in [m],
or raises [Not_found] if no such binding exists.
- NB in Coq, the exception mechanism becomes a option type. *)
+ NB: in Coq, the exception mechanism becomes a option type. *)
- Parameter remove key -> t elt -> t elt.
+ Parameter remove : key -> t elt -> t elt.
(** [remove x m] returns a map containing the same bindings as [m],
except for [x] which is unbound in the returned map. *)
- Parameter mem key -> t elt -> bool.
+ Parameter mem : key -> t elt -> bool.
(** [mem x m] returns [true] if [m] contains a binding for [x],
and [false] otherwise. *)
- (** Coq comment [iter] is useless in a purely functional world *)
- (** val iter (key -> 'a -> unit) -> 'a t -> unit *)
+ (** Coq comment: [iter] is useless in a purely functional world *)
+ (** val iter : (key -> 'a -> unit) -> 'a t -> unit *)
(** iter f m applies f to all bindings in map m. f receives the key as
first argument, and the associated value as second argument.
The bindings are passed to f in increasing order with respect to the
ordering over the type of the keys. Only current bindings are
- presented to f bindings hidden by more recent bindings are not
+ presented to f: bindings hidden by more recent bindings are not
passed to f. *)
- Variable elt' Set.
- Variable elt'' Set.
+ Variable elt' : Set.
+ Variable elt'': Set.
- Parameter map (elt -> elt') -> t elt -> t elt'.
+ Parameter map : (elt -> elt') -> t elt -> t elt'.
(** [map f m] returns a map with same domain as [m], where the associated
value a of all bindings of [m] has been replaced by the result of the
application of [f] to [a]. The bindings are passed to [f] in
increasing order with respect to the ordering over the type of the
keys. *)
- Parameter mapi (key -> elt -> elt') -> t elt -> t elt'.
+ Parameter mapi : (key -> elt -> elt') -> t elt -> t elt'.
(** Same as [S.map], but the function receives as arguments both the
key and the associated value for each binding of the map. *)
- Parameter map2 (option elt -> option elt' -> option elt'') -> t elt -> t elt' -> t elt''.
+ Parameter map2 : (option elt -> option elt' -> option elt'') -> t elt -> t elt' -> t elt''.
(** Not present in Ocaml.
[map f m m'] creates a new map whose bindings belong to the ones of either
[m] or [m']. The presence and value for a key [k] is determined by [f e e']
where [e] and [e'] are the (optional) bindings of [k] in [m] and [m']. *)
- Parameter elements t elt -> list (key*elt).
+ Parameter elements : t elt -> list (key*elt).
(** Not present in Ocaml.
[elements m] returns an assoc list corresponding to the bindings of [m].
Elements of this list are sorted with respect to their first components.
Useful to specify [fold] ... *)
- Parameter fold forall A: Set, (key -> elt -> A -> A) -> t elt -> A -> A.
+ Parameter fold : forall A: Set, (key -> elt -> A -> A) -> t elt -> A -> A.
(** [fold f m a] computes [(f kN dN ... (f k1 d1 a)...)],
where [k1] ... [kN] are the keys of all bindings in [m]
(in increasing order), and [d1] ... [dN] are the associated data. *)
- Parameter equal (elt -> elt -> bool) -> t elt -> t elt -> bool.
+ Parameter equal : (elt -> elt -> bool) -> t elt -> t elt -> bool.
(** [equal cmp m1 m2] tests whether the maps [m1] and [m2] are equal,
that is, contain equal keys and associate them with equal data.
[cmp] is the equality predicate used to compare the data associated
@@ -111,97 +111,97 @@ Module Type S.
Section Spec.
- Variable m m' m'' t elt.
- Variable x y z key.
- Variable e e' elt.
+ Variable m m' m'' : t elt.
+ Variable x y z : key.
+ Variable e e' : elt.
- Parameter MapsTo key -> elt -> t elt -> Prop.
+ Parameter MapsTo : key -> elt -> t elt -> Prop.
- Definition In (kkey)(m: t elt) : Prop := exists e:elt, MapsTo k e m.
+ Definition In (k:key)(m: t elt) : Prop := exists e:elt, MapsTo k e m.
- Definition Empty m = forall (a : key)(e:elt) , ~ MapsTo a e m.
+ Definition Empty m := forall (a : key)(e:elt) , ~ MapsTo a e m.
- Definition eq_key (p p'key*elt) := E.eq (fst p) (fst p').
+ Definition eq_key (p p':key*elt) := E.eq (fst p) (fst p').
- Definition eq_key_elt (p p'key*elt) :=
+ Definition eq_key_elt (p p':key*elt) :=
E.eq (fst p) (fst p') /\ (snd p) = (snd p').
- Definition lt_key (p p'key*elt) := E.lt (fst p) (fst p').
+ Definition lt_key (p p':key*elt) := E.lt (fst p) (fst p').
(** Specification of [MapsTo] *)
- Parameter MapsTo_1 E.eq x y -> MapsTo x e m -> MapsTo y e m.
+ Parameter MapsTo_1 : E.eq x y -> MapsTo x e m -> MapsTo y e m.
(** Specification of [mem] *)
- Parameter mem_1 In x m -> mem x m = true.
- Parameter mem_2 mem x m = true -> In x m.
+ Parameter mem_1 : In x m -> mem x m = true.
+ Parameter mem_2 : mem x m = true -> In x m.
(** Specification of [empty] *)
- Parameter empty_1 Empty empty.
+ Parameter empty_1 : Empty empty.
(** Specification of [is_empty] *)
- Parameter is_empty_1 Empty m -> is_empty m = true.
- Parameter is_empty_2 is_empty m = true -> Empty m.
+ Parameter is_empty_1 : Empty m -> is_empty m = true.
+ Parameter is_empty_2 : is_empty m = true -> Empty m.
(** Specification of [add] *)
- Parameter add_1 E.eq x y -> MapsTo y e (add x e m).
- Parameter add_2 ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
- Parameter add_3 ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
+ Parameter add_1 : E.eq x y -> MapsTo y e (add x e m).
+ Parameter add_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
+ Parameter add_3 : ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
(** Specification of [remove] *)
- Parameter remove_1 E.eq x y -> ~ In y (remove x m).
- Parameter remove_2 ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
- Parameter remove_3 MapsTo y e (remove x m) -> MapsTo y e m.
+ Parameter remove_1 : E.eq x y -> ~ In y (remove x m).
+ Parameter remove_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
+ Parameter remove_3 : MapsTo y e (remove x m) -> MapsTo y e m.
(** Specification of [find] *)
- Parameter find_1 MapsTo x e m -> find x m = Some e.
- Parameter find_2 find x m = Some e -> MapsTo x e m.
+ Parameter find_1 : MapsTo x e m -> find x m = Some e.
+ Parameter find_2 : find x m = Some e -> MapsTo x e m.
(** Specification of [elements] *)
- Parameter elements_1
+ Parameter elements_1 :
MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
- Parameter elements_2
+ Parameter elements_2 :
InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
- Parameter elements_3 sort lt_key (elements m).
+ Parameter elements_3 : sort lt_key (elements m).
(** Specification of [fold] *)
- Parameter fold_1
- forall (A Set) (i : A) (f : key -> elt -> A -> A),
+ Parameter fold_1 :
+ forall (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.
- Definition Equal cmp m m' =
+ Definition Equal cmp 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).
- Variable cmp elt -> elt -> bool.
+ Variable cmp : elt -> elt -> bool.
(** Specification of [equal] *)
- Parameter equal_1 Equal cmp m m' -> equal cmp m m' = true.
- Parameter equal_2 equal cmp m m' = true -> Equal cmp m m'.
+ Parameter equal_1 : Equal cmp m m' -> equal cmp m m' = true.
+ Parameter equal_2 : equal cmp m m' = true -> Equal cmp m m'.
End Spec.
End Types.
(** Specification of [map] *)
- Parameter map_1 forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:elt->elt'),
+ Parameter 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).
- Parameter map_2 forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'),
+ Parameter map_2 : forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'),
In x (map f m) -> In x m.
(** Specification of [mapi] *)
- Parameter mapi_1 forall (elt elt':Set)(m: t elt)(x:key)(e:elt)
- (fkey->elt->elt'), MapsTo x e m ->
+ Parameter 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).
- Parameter mapi_2 forall (elt elt':Set)(m: t elt)(x:key)
- (fkey->elt->elt'), In x (mapi f m) -> In x m.
+ Parameter mapi_2 : forall (elt elt':Set)(m: t elt)(x:key)
+ (f:key->elt->elt'), In x (mapi f m) -> In x m.
(** Specification of [map2] *)
- Parameter map2_1 forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
- (xkey)(f:option elt->option elt'->option elt''),
+ Parameter 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').
- Parameter map2_2 forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
- (xkey)(f:option elt->option elt'->option elt''),
+ Parameter 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'.
Hint Immediate MapsTo_1 mem_2 is_empty_2.
@@ -214,28 +214,28 @@ End S.
Module Type Sord.
- Declare Module Data OrderedType.
- Declare Module MapS S.
+ Declare Module Data : OrderedType.
+ Declare Module MapS : S.
Import MapS.
- Definition t = MapS.t Data.t.
+ Definition t := MapS.t Data.t.
- Parameter eq t -> t -> Prop.
- Parameter lt t -> t -> Prop.
+ Parameter eq : t -> t -> Prop.
+ Parameter lt : t -> t -> Prop.
- Axiom eq_refl forall m : t, eq m m.
- Axiom eq_sym forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
- Axiom eq_trans forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
- Axiom lt_trans forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
- Axiom lt_not_eq forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
+ Axiom eq_refl : forall m : t, eq m m.
+ Axiom eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
+ Axiom eq_trans : forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
+ Axiom lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
+ Axiom lt_not_eq : forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
- Definition cmp e e' = match Data.compare e e' with Eq _ => true | _ => false end.
+ Definition cmp e e' := match Data.compare e e' with Eq _ => true | _ => false end.
- Parameter eq_1 forall m m', Equal cmp m m' -> eq m m'.
- Parameter eq_2 forall m m', eq m m' -> Equal cmp m m'.
+ Parameter eq_1 : forall m m', Equal cmp m m' -> eq m m'.
+ Parameter eq_2 : forall m m', eq m m' -> Equal cmp m m'.
- Parameter compare forall m1 m2, Compare lt eq m1 m2.
- (** Total ordering between maps. The first argument (in Coq Data.compare)
+ Parameter compare : forall m1 m2, Compare lt eq m1 m2.
+ (** Total ordering between maps. The first argument (in Coq: Data.compare)
is a total ordering used to compare data associated with equal keys
in the two maps. *)
diff --git a/theories/FSets/FMapList.v b/theories/FSets/FMapList.v
index be6d56e5c..a2005c1fe 100644
--- a/theories/FSets/FMapList.v
+++ b/theories/FSets/FMapList.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetList.v,v 1.12 2006/03/10 10:49:48 letouzey Exp $ *)
(** * Finite map library *)
@@ -22,44 +22,44 @@ Unset Strict Implicit.
Arguments Scope list [type_scope].
-Module Raw (XOrderedType).
+Module Raw (X:OrderedType).
-Module E = X.
-Module MX = OrderedTypeFacts X.
-Module PX = PairOrderedType X.
+Module E := X.
+Module MX := OrderedTypeFacts X.
+Module PX := PairOrderedType X.
Import MX.
Import PX.
-Definition key = X.t.
-Definition t (eltSet) := list (X.t * elt).
+Definition key := X.t.
+Definition t (elt:Set) := list (X.t * elt).
Section Elt.
-Variable elt Set.
+Variable elt : Set.
-(* Now in PairOrderedtype
-Definition eqk (p p'key*elt) := X.eq (fst p) (fst p').
-Definition eqke (p p'key*elt) :=
+(* Now in PairOrderedtype:
+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').
-Definition ltk (p p'key*elt) := X.lt (fst p) (fst p').
-Definition MapsTo (kkey)(e:elt):= InA eqke (k,e).
-Definition In k m = exists e:elt, MapsTo k e m.
+Definition ltk (p p':key*elt) := X.lt (fst p) (fst p').
+Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
+Definition In k m := exists e:elt, MapsTo k e m.
*)
-Notation eqk = (eqk (elt:=elt)).
-Notation eqke = (eqke (elt:=elt)).
-Notation ltk = (ltk (elt:=elt)).
-Notation MapsTo = (MapsTo (elt:=elt)).
-Notation In = (In (elt:=elt)).
-Notation Sort = (sort ltk).
-Notation Inf = (lelistA (ltk)).
+Notation eqk := (eqk (elt:=elt)).
+Notation eqke := (eqke (elt:=elt)).
+Notation ltk := (ltk (elt:=elt)).
+Notation MapsTo := (MapsTo (elt:=elt)).
+Notation In := (In (elt:=elt)).
+Notation Sort := (sort ltk).
+Notation Inf := (lelistA (ltk)).
(** * [empty] *)
-Definition empty t elt := nil.
+Definition empty : t elt := nil.
-Definition Empty m = forall (a : key)(e:elt) , ~ MapsTo a e m.
+Definition Empty m := forall (a : key)(e:elt) , ~ MapsTo a e m.
-Lemma empty_1 Empty empty.
+Lemma empty_1 : Empty empty.
Proof.
unfold Empty,empty.
intros a e.
@@ -68,25 +68,25 @@ Proof.
Qed.
Hint Resolve empty_1.
-Lemma empty_sorted Sort empty.
+Lemma empty_sorted : Sort empty.
Proof.
unfold empty; auto.
Qed.
(** * [is_empty] *)
-Definition is_empty (l t elt) : bool := if l then true else false.
+Definition is_empty (l : t elt) : bool := if l then true else false.
-Lemma is_empty_1 forall m, Empty m -> is_empty m = true.
+Lemma is_empty_1 :forall m, Empty m -> is_empty m = true.
Proof.
unfold Empty, PX.MapsTo.
intros m.
case m;auto.
intros (k,e) l inlist.
- absurd (InA eqke (k, e) ((k, e) : l));auto.
+ absurd (InA eqke (k, e) ((k, e) :: l));auto.
Qed.
-Lemma is_empty_2 forall m, is_empty m = true -> Empty m.
+Lemma is_empty_2 : forall m, is_empty m = true -> Empty m.
Proof.
intros m.
case m;auto.
@@ -96,10 +96,10 @@ Qed.
(** * [mem] *)
-Fixpoint mem (k key) (s : t elt) {struct s} : bool :=
+Fixpoint mem (k : key) (s : t elt) {struct s} : bool :=
match s with
| nil => false
- | (k',_) : l =>
+ | (k',_) :: l =>
match X.compare k k' with
| Lt _ => false
| Eq _ => true
@@ -107,14 +107,14 @@ Fixpoint mem (k key) (s : t elt) {struct s} : bool :=
end
end.
-Lemma mem_1 forall m (Hm:Sort m) x, In x m -> mem x m = true.
+Lemma mem_1 : forall m (Hm:Sort m) x, In x m -> mem x m = true.
Proof.
intros m Hm x; generalize Hm; clear Hm.
functional induction mem x m;intros sorted belong1;trivial.
inversion belong1. inversion H.
- absurd (In k ((k', e) : l));try assumption.
+ absurd (In k ((k', e) :: l));try assumption.
apply Sort_Inf_NotIn with e;auto.
apply H.
@@ -124,7 +124,7 @@ Proof.
absurd (X.eq k k');auto.
Qed.
-Lemma mem_2 forall m (Hm:Sort m) x, mem x m = true -> In x m.
+Lemma mem_2 : forall m (Hm:Sort 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 sorted hyp;try ((inversion hyp);fail).
@@ -136,10 +136,10 @@ Qed.
(** * [find] *)
-Fixpoint find (kkey) (s: t elt) {struct s} : option elt :=
+Fixpoint find (k:key) (s: t elt) {struct s} : option elt :=
match s with
| nil => None
- | (k',x):s' =>
+ | (k',x)::s' =>
match X.compare k k' with
| Lt _ => None
| Eq _ => Some x
@@ -147,13 +147,13 @@ Fixpoint find (kkey) (s: t elt) {struct s} : option elt :=
end
end.
-Lemma find_2 forall m x e, find x m = Some e -> MapsTo x e m.
+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.
Qed.
-Lemma find_1 forall m (Hm:Sort m) x e, MapsTo x e m -> find x m = Some e.
+Lemma find_1 : forall m (Hm:Sort 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.
@@ -174,25 +174,25 @@ Qed.
(** * [add] *)
-Fixpoint add (k key) (x : elt) (s : t elt) {struct s} : t elt :=
+Fixpoint add (k : key) (x : elt) (s : t elt) {struct s} : t elt :=
match s with
- | nil => (k,x) : nil
- | (k',y) : l =>
+ | nil => (k,x) :: nil
+ | (k',y) :: l =>
match X.compare k k' with
- | Lt _ => (k,x):s
- | Eq _ => (k,x):l
- | Gt _ => (k',y) : add k x l
+ | Lt _ => (k,x)::s
+ | Eq _ => (k,x)::l
+ | Gt _ => (k',y) :: add k x l
end
end.
-Lemma add_1 forall m x y e, X.eq x y -> MapsTo y e (add x e m).
+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.
Qed.
-Lemma add_2 forall m x y e e',
+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'.
@@ -205,7 +205,7 @@ Proof.
intros y' e' eqky'; inversion_clear 1; intuition.
Qed.
-Lemma add_3 forall m x y e e',
+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.
@@ -216,7 +216,7 @@ Proof.
inversion_clear H1; auto.
Qed.
-Lemma add_Inf forall (m:t elt)(x x':key)(e e':elt),
+Lemma add_Inf : forall (m:t elt)(x x':key)(e e':elt),
Inf (x',e') m -> ltk (x',e') (x,e) -> Inf (x',e') (add x e m).
Proof.
induction m.
@@ -229,7 +229,7 @@ Proof.
Qed.
Hint Resolve add_Inf.
-Lemma add_sorted forall m (Hm:Sort m) x e, Sort (add x e m).
+Lemma add_sorted : forall m (Hm:Sort m) x e, Sort (add x e m).
Proof.
induction m.
simpl; intuition.
@@ -242,18 +242,18 @@ Qed.
(** * [remove] *)
-Fixpoint remove (k key) (s : t elt) {struct s} : t elt :=
+Fixpoint remove (k : key) (s : t elt) {struct s} : t elt :=
match s with
| nil => nil
- | (k',x) : l =>
+ | (k',x) :: l =>
match X.compare k k' with
| Lt _ => s
| Eq _ => l
- | Gt _ => (k',x) : remove k l
+ | Gt _ => (k',x) :: remove k l
end
end.
-Lemma remove_1 forall m (Hm:Sort m) x y, X.eq x y -> ~ In y (remove x m).
+Lemma remove_1 : forall m (Hm:Sort 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;subst;try clear H_eq_1.
@@ -268,14 +268,14 @@ Proof.
apply Inf_eq with (k',x);auto; compute; apply X.eq_trans with k; auto.
inversion_clear Hm.
- assert (notin~ In y (remove k l)) by auto.
+ assert (notin:~ In y (remove k l)) by auto.
intros (x0,abs).
inversion_clear abs.
compute in H3; destruct H3; order.
apply notin; exists x0; auto.
Qed.
-Lemma remove_2 forall m (Hm:Sort m) x y e,
+Lemma remove_2 : forall m (Hm:Sort 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.
@@ -286,7 +286,7 @@ Proof.
inversion_clear 1; inversion_clear 2; auto.
Qed.
-Lemma remove_3 forall m (Hm:Sort m) x y e,
+Lemma remove_3 : forall m (Hm:Sort 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.
@@ -294,7 +294,7 @@ Proof.
inversion_clear 1; inversion_clear 1; auto.
Qed.
-Lemma remove_Inf forall (m:t elt)(Hm : Sort m)(x x':key)(e':elt),
+Lemma remove_Inf : forall (m:t elt)(Hm : Sort m)(x x':key)(e':elt),
Inf (x',e') m -> Inf (x',e') (remove x m).
Proof.
induction m.
@@ -309,7 +309,7 @@ Proof.
Qed.
Hint Resolve remove_Inf.
-Lemma remove_sorted forall m (Hm:Sort m) x, Sort (remove x m).
+Lemma remove_sorted : forall m (Hm:Sort m) x, Sort (remove x m).
Proof.
induction m.
simpl; intuition.
@@ -320,35 +320,35 @@ Qed.
(** * [elements] *)
-Definition elements (m t elt) := m.
+Definition elements (m: t elt) := m.
-Lemma elements_1 forall m x e,
+Lemma elements_1 : forall m x e,
MapsTo x e m -> InA eqke (x,e) (elements m).
Proof.
auto.
Qed.
-Lemma elements_2 forall m x e,
+Lemma elements_2 : forall m x e,
InA eqke (x,e) (elements m) -> MapsTo x e m.
Proof.
auto.
Qed.
-Lemma elements_3 forall m (Hm:Sort m), sort ltk (elements m).
+Lemma elements_3 : forall m (Hm:Sort m), sort ltk (elements m).
Proof.
auto.
Qed.
(** * [fold] *)
-Fixpoint fold (ASet)(f:key->elt->A->A)(m:t elt) {struct m} : A -> A :=
+Fixpoint fold (A:Set)(f:key->elt->A->A)(m:t elt) {struct m} : A -> A :=
fun acc =>
match m with
| nil => acc
- | (k,e):m' => fold f m' (f k e acc)
+ | (k,e)::m' => fold f m' (f k e acc)
end.
-Lemma fold_1 forall m (A:Set)(i:A)(f:key->elt->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.
@@ -356,10 +356,10 @@ Qed.
(** * [equal] *)
-Fixpoint equal (cmpelt->elt->bool)(m m' : t elt) { struct m } : bool :=
+Fixpoint equal (cmp:elt->elt->bool)(m m' : t elt) { struct m } : bool :=
match m, m' with
| nil, nil => true
- | (x,e):l, (x',e')::l' =>
+ | (x,e)::l, (x',e')::l' =>
match X.compare x x' with
| Eq _ => cmp e e' && equal cmp l l'
| _ => false
@@ -367,11 +367,11 @@ Fixpoint equal (cmpelt->elt->bool)(m m' : t elt) { struct m } : bool :=
| _, _ => false
end.
-Definition Equal cmp m m' =
+Definition Equal cmp 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).
-Lemma equal_1 forall m (Hm:Sort m) m' (Hm': Sort m') cmp,
+Lemma equal_1 : forall m (Hm:Sort m) m' (Hm': Sort m') cmp,
Equal cmp m m' -> equal cmp m m' = true.
Proof.
intros m Hm m' Hm' cmp; generalize Hm Hm'; clear Hm Hm'.
@@ -390,7 +390,7 @@ Proof.
inversion H.
destruct (H0 x).
- assert (In x ((x',e'):l')).
+ assert (In x ((x',e')::l')).
apply H; auto.
exists e; auto.
destruct (In_inv H3).
@@ -408,7 +408,7 @@ Proof.
inversion_clear Hm'; auto.
unfold Equal; intuition.
destruct (H1 k).
- assert (In k ((x,e) :l)).
+ assert (In k ((x,e) ::l)).
destruct H3 as (e'', hyp); exists e''; auto.
destruct (In_inv (H4 H6)); auto.
inversion_clear Hm.
@@ -416,7 +416,7 @@ Proof.
destruct H3 as (e'', hyp); exists e''; auto.
apply MapsTo_eq with k; auto; order.
destruct (H1 k).
- assert (In k ((x',e') :l')).
+ assert (In k ((x',e') ::l')).
destruct H3 as (e'', hyp); exists e''; auto.
destruct (In_inv (H5 H6)); auto.
inversion_clear Hm'.
@@ -426,7 +426,7 @@ Proof.
apply H2 with k; destruct (eq_dec x k); auto.
destruct (H0 x').
- assert (In x' ((x,e):l)).
+ assert (In x' ((x,e)::l)).
apply H2; auto.
exists e'; auto.
destruct (In_inv H3).
@@ -437,7 +437,7 @@ Proof.
elim (Sort_Inf_NotIn H5 H7 H4).
Qed.
-Lemma equal_2 forall m (Hm:Sort m) m' (Hm:Sort m') cmp,
+Lemma equal_2 : forall m (Hm:Sort m) m' (Hm:Sort m') cmp,
equal cmp m m' = true -> Equal cmp m m'.
Proof.
intros m Hm m' Hm' cmp; generalize Hm Hm'; clear Hm Hm'.
@@ -476,9 +476,9 @@ Qed.
(** This lemma isn't part of the spec of [Equal], but is used in [FMapAVL] *)
-Lemma equal_cons forall cmp l1 l2 x y, Sort (x::l1) -> Sort (y::l2) ->
+Lemma equal_cons : forall cmp l1 l2 x y, Sort (x::l1) -> Sort (y::l2) ->
eqk x y -> cmp (snd x) (snd y) = true ->
- (Equal cmp l1 l2 <-> Equal cmp (x : l1) (y :: l2)).
+ (Equal cmp l1 l2 <-> Equal cmp (x :: l1) (y :: l2)).
Proof.
intros.
inversion H; subst.
@@ -497,20 +497,20 @@ Proof.
rewrite H2; simpl; auto.
Qed.
-Variable elt'Set.
+Variable elt':Set.
(** * [map] and [mapi] *)
-Fixpoint map (felt -> elt') (m:t elt) {struct m} : t elt' :=
+Fixpoint map (f:elt -> elt') (m:t elt) {struct m} : t elt' :=
match m with
| nil => nil
- | (k,e):m' => (k,f e) :: map f m'
+ | (k,e)::m' => (k,f e) :: map f m'
end.
-Fixpoint mapi (f key -> elt -> elt') (m:t elt) {struct m} : t elt' :=
+Fixpoint mapi (f: key -> elt -> elt') (m:t elt) {struct m} : t elt' :=
match m with
| nil => nil
- | (k,e):m' => (k,f k e) :: mapi f m'
+ | (k,e)::m' => (k,f k e) :: mapi f m'
end.
End Elt.
@@ -518,11 +518,11 @@ Section Elt2.
(* A new section is necessary for previous definitions to work
with different [elt], especially [MapsTo]... *)
-Variable elt elt' Set.
+Variable elt elt' : Set.
(** Specification of [map] *)
-Lemma map_1 forall (m:t elt)(x:key)(e:elt)(f:elt->elt'),
+Lemma map_1 : forall (m:t elt)(x:key)(e:elt)(f:elt->elt'),
MapsTo x e m -> MapsTo x (f e) (map f m).
Proof.
intros m x e f.
@@ -539,7 +539,7 @@ Proof.
unfold MapsTo in *; auto.
Qed.
-Lemma map_2 forall (m:t elt)(x:key)(f:elt->elt'),
+Lemma map_2 : forall (m:t elt)(x:key)(f:elt->elt'),
In x (map f m) -> In x m.
Proof.
intros m x f.
@@ -560,7 +560,7 @@ Proof.
constructor 2; auto.
Qed.
-Lemma map_lelistA forall (m: t elt)(x:key)(e:elt)(e':elt')(f:elt->elt'),
+Lemma map_lelistA : forall (m: t elt)(x:key)(e:elt)(e':elt')(f:elt->elt'),
lelistA (@ltk elt) (x,e) m ->
lelistA (@ltk elt') (x,e') (map f m).
Proof.
@@ -572,7 +572,7 @@ Qed.
Hint Resolve map_lelistA.
-Lemma map_sorted forall (m: t elt)(Hm : sort (@ltk elt) m)(f:elt -> elt'),
+Lemma map_sorted : forall (m: t elt)(Hm : sort (@ltk elt) m)(f:elt -> elt'),
sort (@ltk elt') (map f m).
Proof.
induction m; simpl; auto.
@@ -585,7 +585,7 @@ Qed.
(** Specification of [mapi] *)
-Lemma mapi_1 forall (m:t elt)(x:key)(e:elt)(f:key->elt->elt'),
+Lemma mapi_1 : forall (m:t elt)(x:key)(e:elt)(f:key->elt->elt'),
MapsTo x e m ->
exists y, X.eq y x /\ MapsTo x (f y e) (mapi f m).
Proof.
@@ -607,7 +607,7 @@ Proof.
Qed.
-Lemma mapi_2 forall (m:t elt)(x:key)(f:key->elt->elt'),
+Lemma mapi_2 : forall (m:t elt)(x:key)(f:key->elt->elt'),
In x (mapi f m) -> In x m.
Proof.
intros m x f.
@@ -628,7 +628,7 @@ Proof.
constructor 2; auto.
Qed.
-Lemma mapi_lelistA forall (m: t elt)(x:key)(e:elt)(f:key->elt->elt'),
+Lemma mapi_lelistA : forall (m: t elt)(x:key)(e:elt)(f:key->elt->elt'),
lelistA (@ltk elt) (x,e) m ->
lelistA (@ltk elt') (x,f x e) (mapi f m).
Proof.
@@ -640,7 +640,7 @@ Qed.
Hint Resolve mapi_lelistA.
-Lemma mapi_sorted forall m (Hm : sort (@ltk elt) m)(f: key ->elt -> elt'),
+Lemma mapi_sorted : forall m (Hm : sort (@ltk elt) m)(f: key ->elt -> elt'),
sort (@ltk elt') (mapi f m).
Proof.
induction m; simpl; auto.
@@ -654,35 +654,35 @@ Section Elt3.
(** * [map2] *)
-Variable elt elt' elt'' Set.
-Variable f option elt -> option elt' -> option elt''.
+Variable elt elt' elt'' : Set.
+Variable f : option elt -> option elt' -> option elt''.
-Definition option_cons (ASet)(k:key)(o:option A)(l:list (key*A)) :=
+Definition option_cons (A:Set)(k:key)(o:option A)(l:list (key*A)) :=
match o with
- | Some e => (k,e):l
+ | Some e => (k,e)::l
| None => l
end.
-Fixpoint map2_l (m t elt) : t elt'' :=
+Fixpoint map2_l (m : t elt) : t elt'' :=
match m with
| nil => nil
- | (k,e):l => option_cons k (f (Some e) None) (map2_l l)
+ | (k,e)::l => option_cons k (f (Some e) None) (map2_l l)
end.
-Fixpoint map2_r (m' t elt') : t elt'' :=
+Fixpoint map2_r (m' : t elt') : t elt'' :=
match m' with
| nil => nil
- | (k,e'):l' => option_cons k (f None (Some e')) (map2_r l')
+ | (k,e')::l' => option_cons k (f None (Some e')) (map2_r l')
end.
-Fixpoint map2 (m t elt) : t elt' -> t elt'' :=
+Fixpoint map2 (m : t elt) : t elt' -> t elt'' :=
match m with
| nil => map2_r
- | (k,e) : l =>
- fix map2_aux (m' t elt') : t elt'' :=
+ | (k,e) :: l =>
+ fix map2_aux (m' : t elt') : t elt'' :=
match m' with
| nil => map2_l m
- | (k',e') : l' =>
+ | (k',e') :: l' =>
match X.compare k k' with
| Lt _ => option_cons k (f (Some e) None) (map2 l m')
| Eq _ => option_cons k (f (Some e) (Some e')) (map2 l l')
@@ -691,33 +691,33 @@ Fixpoint map2 (m t elt) : t elt' -> t elt'' :=
end
end.
-Notation oee' = (option elt * option elt')%type.
+Notation oee' := (option elt * option elt')%type.
-Fixpoint combine (m t elt) : t elt' -> t oee' :=
+Fixpoint combine (m : t elt) : t elt' -> t oee' :=
match m with
| nil => map (fun e' => (None,Some e'))
- | (k,e) : l =>
- fix combine_aux (m't elt') : list (key * oee') :=
+ | (k,e) :: l =>
+ fix combine_aux (m':t elt') : list (key * oee') :=
match m' with
| nil => map (fun e => (Some e,None)) m
- | (k',e') : l' =>
+ | (k',e') :: l' =>
match X.compare k k' with
- | Lt _ => (k,(Some e, None)):combine l m'
- | Eq _ => (k,(Some e, Some e')):combine l l'
- | Gt _ => (k',(None,Some e')):combine_aux l'
+ | Lt _ => (k,(Some e, None))::combine l m'
+ | Eq _ => (k,(Some e, Some e'))::combine l l'
+ | Gt _ => (k',(None,Some e'))::combine_aux l'
end
end
end.
-Definition fold_right_pair (A B CSet)(f: A->B->C->C)(l:list (A*B))(i:C) :=
+Definition fold_right_pair (A B C:Set)(f: A->B->C->C)(l:list (A*B))(i:C) :=
List.fold_right (fun p => f (fst p) (snd p)) i l.
-Definition map2_alt m m' =
- let m0 t oee' := combine m m' in
- let m1 t (option elt'') := map (fun p => f (fst p) (snd p)) m0 in
- fold_right_pair (option_cons (A=elt'')) m1 nil.
+Definition map2_alt m m' :=
+ let m0 : t oee' := combine m m' in
+ let m1 : t (option elt'') := map (fun p => f (fst p) (snd p)) m0 in
+ fold_right_pair (option_cons (A:=elt'')) m1 nil.
-Lemma map2_alt_equiv forall m m', map2_alt m m' = map2 m m'.
+Lemma map2_alt_equiv : forall m m', map2_alt m m' = map2 m m'.
Proof.
unfold map2_alt.
induction m.
@@ -741,8 +741,8 @@ Proof.
apply IHm'.
Qed.
-Lemma combine_lelistA
- forall m m' (xkey)(e:elt)(e':elt')(e'':oee'),
+Lemma combine_lelistA :
+ forall m m' (x:key)(e:elt)(e':elt')(e'':oee'),
lelistA (@ltk elt) (x,e) m ->
lelistA (@ltk elt') (x,e') m' ->
lelistA (@ltk oee') (x,e'') (combine m m').
@@ -754,8 +754,8 @@ Proof.
induction m'.
intros.
destruct a.
- replace (combine ((t0, e0) : m) nil) with
- (map (fun e => (Some e,None (A=elt'))) ((t0,e0)::m)); auto.
+ replace (combine ((t0, e0) :: m) nil) with
+ (map (fun e => (Some e,None (A:=elt'))) ((t0,e0)::m)); auto.
exact (map_lelistA _ _ H).
intros.
simpl.
@@ -767,8 +767,8 @@ Proof.
Qed.
Hint Resolve combine_lelistA.
-Lemma combine_sorted
- forall m (Hm sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m'),
+Lemma combine_sorted :
+ forall m (Hm : sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m'),
sort (@ltk oee') (combine m m').
Proof.
induction m.
@@ -778,8 +778,8 @@ Proof.
induction m'.
intros; clear Hm'.
destruct a.
- replace (combine ((t0, e) : m) nil) with
- (map (fun e => (Some e,None (A=elt'))) ((t0,e)::m)); auto.
+ replace (combine ((t0, e) :: m) nil) with
+ (map (fun e => (Some e,None (A:=elt'))) ((t0,e)::m)); auto.
apply map_sorted; auto.
intros.
simpl.
@@ -787,32 +787,32 @@ Proof.
destruct (X.compare k k').
inversion_clear Hm.
constructor; auto.
- assert (lelistA (ltk (elt=elt')) (k, e') ((k',e')::m')) by auto.
+ assert (lelistA (ltk (elt:=elt')) (k, e') ((k',e')::m')) by auto.
exact (combine_lelistA _ H0 H1).
inversion_clear Hm; inversion_clear Hm'.
constructor; auto.
- assert (lelistA (ltk (elt=elt')) (k, e') m') by apply Inf_eq with (k',e'); auto.
+ assert (lelistA (ltk (elt:=elt')) (k, e') m') by apply Inf_eq with (k',e'); auto.
exact (combine_lelistA _ H0 H3).
inversion_clear Hm; inversion_clear Hm'.
constructor; auto.
- change (lelistA (ltk (elt=oee')) (k', (None, Some e'))
- (combine ((k,e):m) m')).
- assert (lelistA (ltk (elt=elt)) (k', e) ((k,e)::m)) by auto.
+ change (lelistA (ltk (elt:=oee')) (k', (None, Some e'))
+ (combine ((k,e)::m) m')).
+ assert (lelistA (ltk (elt:=elt)) (k', e) ((k,e)::m)) by auto.
exact (combine_lelistA _ H3 H2).
Qed.
-Lemma map2_sorted
- forall m (Hm sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m'),
+Lemma map2_sorted :
+ forall m (Hm : sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m'),
sort (@ltk elt'') (map2 m m').
Proof.
intros.
rewrite <- map2_alt_equiv.
unfold map2_alt.
- assert (H0=combine_sorted Hm Hm').
- set (l0=combine m m') in *; clearbody l0.
- set (f'= fun p : oee' => f (fst p) (snd p)).
- assert (H1=map_sorted (elt' := option elt'') H0 f').
- set (l1=map f' l0) in *; clearbody l1.
+ assert (H0:=combine_sorted Hm Hm').
+ set (l0:=combine m m') in *; clearbody l0.
+ set (f':= fun p : oee' => f (fst p) (snd p)).
+ assert (H1:=map_sorted (elt' := option elt'') H0 f').
+ set (l1:=map f' l0) in *; clearbody l1.
clear f' f H0 l0 Hm Hm' m m'.
induction l1.
simpl; auto.
@@ -829,17 +829,17 @@ Proof.
red in H1; simpl in H1.
inversion_clear H.
apply IHl1; auto.
- apply Inf_lt with (t1, None (A=elt'')); auto.
+ apply Inf_lt with (t1, None (A:=elt'')); auto.
Qed.
-Definition at_least_one (ooption elt)(o':option elt') :=
+Definition at_least_one (o:option elt)(o':option elt') :=
match o, o' with
| None, None => None
| _, _ => Some (o,o')
end.
-Lemma combine_1
- forall m (Hm sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m') (x:key),
+Lemma combine_1 :
+ forall m (Hm : sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m') (x:key),
find x (combine m m') = at_least_one (find x m) (find x m').
Proof.
induction m.
@@ -869,37 +869,37 @@ Proof.
rewrite IHm; auto; simpl; elim_comp; auto.
rewrite IHm; auto; simpl; elim_comp; auto.
rewrite IHm; auto; simpl; elim_comp; auto.
- change (find x (combine ((k, e) : m) m') = at_least_one None (find x m')).
+ change (find x (combine ((k, e) :: m) m') = at_least_one None (find x m')).
rewrite IHm'; auto.
simpl find; elim_comp; auto.
- change (find x (combine ((k, e) : m) m') = Some (Some e, find x m')).
+ change (find x (combine ((k, e) :: m) m') = Some (Some e, find x m')).
rewrite IHm'; auto.
simpl find; elim_comp; auto.
- change (find x (combine ((k, e) : m) m') =
+ change (find x (combine ((k, e) :: m) m') =
at_least_one (find x m) (find x m')).
rewrite IHm'; auto.
simpl find; elim_comp; auto.
Qed.
-Definition at_least_one_then_f (ooption elt)(o':option elt') :=
+Definition at_least_one_then_f (o:option elt)(o':option elt') :=
match o, o' with
| None, None => None
| _, _ => f o o'
end.
-Lemma map2_0
- forall m (Hm sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m') (x:key),
+Lemma map2_0 :
+ forall m (Hm : sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m') (x:key),
find x (map2 m m') = at_least_one_then_f (find x m) (find x m').
Proof.
intros.
rewrite <- map2_alt_equiv.
unfold map2_alt.
- assert (H=combine_1 Hm Hm' x).
- assert (H2=combine_sorted Hm Hm').
- set (f'= fun p : oee' => f (fst p) (snd p)).
- set (m0 = combine m m') in *; clearbody m0.
- set (o=find x m) in *; clearbody o.
- set (o'=find x m') in *; clearbody o'.
+ assert (H:=combine_1 Hm Hm' x).
+ assert (H2:=combine_sorted Hm Hm').
+ set (f':= fun p : oee' => f (fst p) (snd p)).
+ set (m0 := combine m m') in *; clearbody m0.
+ set (o:=find x m) in *; clearbody o.
+ set (o':=find x m') in *; clearbody o'.
clear Hm Hm' m m'.
generalize H; clear H.
match goal with |- ?m=?n -> ?p=?q =>
@@ -916,7 +916,7 @@ Proof.
destruct (IHm0 H0) as (H2,_); apply H2; auto.
rewrite <- H.
case_eq (find x m0); intros; auto.
- assert (ltk (elt=oee') (x,(oo,oo')) (k,(oo,oo'))).
+ assert (ltk (elt:=oee') (x,(oo,oo')) (k,(oo,oo'))).
red; auto.
destruct (Sort_Inf_NotIn H0 (Inf_lt H4 H1)).
exists p; apply find_2; auto.
@@ -929,7 +929,7 @@ Proof.
elim_comp; auto.
destruct (IHm0 H0) as (_,H4); apply H4; auto.
case_eq (find x m0); intros; auto.
- assert (eqk (elt=oee') (k,(oo,oo')) (x,(oo,oo'))).
+ assert (eqk (elt:=oee') (k,(oo,oo')) (x,(oo,oo'))).
red; auto.
destruct (Sort_Inf_NotIn H0 (Inf_eq (eqk_sym H5) H1)).
exists p; apply find_2; auto.
@@ -951,7 +951,7 @@ Proof.
elim_comp; auto.
destruct (IHm0 H0) as (_,H4); apply H4; auto.
case_eq (find x m0); intros; auto.
- assert (ltk (elt=oee') (x,(oo,oo')) (k,(oo,oo'))).
+ assert (ltk (elt:=oee') (x,(oo,oo')) (k,(oo,oo'))).
red; auto.
destruct (Sort_Inf_NotIn H0 (Inf_lt H3 H1)).
exists p; apply find_2; auto.
@@ -967,8 +967,8 @@ Qed.
(** Specification of [map2] *)
-Lemma map2_1
- forall m (Hm sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m')(x:key),
+Lemma map2_1 :
+ forall m (Hm : sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m')(x:key),
In x m \/ In x m' ->
find x (map2 m m') = f (find x m) (find x m').
Proof.
@@ -981,8 +981,8 @@ Proof.
destruct (find x m); simpl; auto.
Qed.
-Lemma map2_2
- forall m (Hm sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m')(x:key),
+Lemma map2_2 :
+ forall m (Hm : sort (@ltk elt) m) m' (Hm' : sort (@ltk elt') m')(x:key),
In x (map2 m m') -> In x m \/ In x m'.
Proof.
intros.
@@ -1002,109 +1002,109 @@ Qed.
End Elt3.
End Raw.
-Module Make (X OrderedType) <: S with Module E := X.
-Module Raw = Raw X.
-Module E = X.
+Module Make (X: OrderedType) <: S with Module E := X.
+Module Raw := Raw X.
+Module E := X.
-Definition key = X.t.
+Definition key := X.t.
-Record slist (eltSet) : Set :=
- {this > Raw.t elt; sorted : sort (@Raw.PX.ltk elt) this}.
-Definition t (eltSet) := slist elt.
+Record slist (elt:Set) : Set :=
+ {this :> Raw.t elt; sorted : sort (@Raw.PX.ltk elt) this}.
+Definition t (elt:Set) := slist elt.
Section Elt.
- Variable elt elt' elt''Set.
-
- Implicit Types m t elt.
-
- Definition empty = Build_slist (Raw.empty_sorted elt).
- Definition is_empty m = Raw.is_empty m.(this).
- Definition add x e m = Build_slist (Raw.add_sorted m.(sorted) x e).
- Definition find x m = Raw.find x m.(this).
- Definition remove x m = Build_slist (Raw.remove_sorted m.(sorted) x).
- Definition mem x m = Raw.mem x m.(this).
- Definition map f m t elt' := Build_slist (Raw.map_sorted m.(sorted) f).
- Definition mapi f m t elt' := Build_slist (Raw.mapi_sorted m.(sorted) f).
- Definition map2 f m (m't elt') : t elt'' :=
+ Variable elt elt' elt'':Set.
+
+ Implicit Types m : t elt.
+
+ Definition empty := Build_slist (Raw.empty_sorted elt).
+ Definition is_empty m := Raw.is_empty m.(this).
+ Definition add x e m := Build_slist (Raw.add_sorted m.(sorted) x e).
+ Definition find x m := Raw.find x m.(this).
+ Definition remove x m := Build_slist (Raw.remove_sorted m.(sorted) x).
+ Definition mem x m := Raw.mem x m.(this).
+ Definition map f m : t elt' := Build_slist (Raw.map_sorted m.(sorted) f).
+ Definition mapi f m : t elt' := Build_slist (Raw.mapi_sorted m.(sorted) f).
+ Definition map2 f m (m':t elt') : t elt'' :=
Build_slist (Raw.map2_sorted f m.(sorted) m'.(sorted)).
- 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 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).
+ 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).
- Definition eq_key = Raw.PX.eqk.
- Definition eq_key_elt = Raw.PX.eqke.
- Definition lt_key = Raw.PX.ltk.
+ Definition eq_key := Raw.PX.eqk.
+ Definition eq_key_elt := Raw.PX.eqke.
+ Definition lt_key := Raw.PX.ltk.
- Definition MapsTo_1 m = @Raw.PX.MapsTo_eq elt m.(this).
+ Definition MapsTo_1 m := @Raw.PX.MapsTo_eq elt m.(this).
- Definition mem_1 m = @Raw.mem_1 elt m.(this) m.(sorted).
- Definition mem_2 m = @Raw.mem_2 elt m.(this) m.(sorted).
+ Definition mem_1 m := @Raw.mem_1 elt m.(this) m.(sorted).
+ Definition mem_2 m := @Raw.mem_2 elt m.(this) m.(sorted).
- Definition empty_1 = @Raw.empty_1.
+ 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 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).
+ 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).
- Definition remove_1 m = @Raw.remove_1 elt m.(this) m.(sorted).
- Definition remove_2 m = @Raw.remove_2 elt m.(this) m.(sorted).
- Definition remove_3 m = @Raw.remove_3 elt m.(this) m.(sorted).
+ Definition remove_1 m := @Raw.remove_1 elt m.(this) m.(sorted).
+ Definition remove_2 m := @Raw.remove_2 elt m.(this) m.(sorted).
+ Definition remove_3 m := @Raw.remove_3 elt m.(this) m.(sorted).
- Definition find_1 m = @Raw.find_1 elt m.(this) m.(sorted).
- Definition find_2 m = @Raw.find_2 elt m.(this).
+ Definition find_1 m := @Raw.find_1 elt m.(this) m.(sorted).
+ 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.(sorted).
+ 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.(sorted).
- Definition fold_1 m = @Raw.fold_1 elt m.(this).
+ 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 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 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 :=
+ Definition map2_1 m (m':t elt') x f :=
@Raw.map2_1 elt elt' elt'' f m.(this) m.(sorted) m'.(this) m'.(sorted) x.
- Definition map2_2 m (m't elt') x f :=
+ Definition map2_2 m (m':t elt') x f :=
@Raw.map2_2 elt elt' elt'' f m.(this) m.(sorted) m'.(this) m'.(sorted) x.
- Definition equal_1 m m' =
+ Definition equal_1 m m' :=
@Raw.equal_1 elt m.(this) m.(sorted) m'.(this) m'.(sorted).
- Definition equal_2 m m' =
+ Definition equal_2 m m' :=
@Raw.equal_2 elt m.(this) m.(sorted) m'.(this) m'.(sorted).
End Elt.
End Make.
-Module Make_ord (X OrderedType)(D : OrderedType) <:
-Sord with Module Data = D
- with Module MapS.E = X.
+Module Make_ord (X: OrderedType)(D : OrderedType) <:
+Sord with Module Data := D
+ with Module MapS.E := X.
-Module Data = D.
-Module MapS = Make(X).
+Module Data := D.
+Module MapS := Make(X).
Import MapS.
-Module MD = OrderedTypeFacts(D).
+Module MD := OrderedTypeFacts(D).
Import MD.
-Definition t = MapS.t D.t.
+Definition t := MapS.t D.t.
-Definition cmp e e' = match D.compare e e' with Eq _ => true | _ => false end.
+Definition cmp e e' := match D.compare e e' with Eq _ => true | _ => false end.
-Fixpoint eq_list (m m' list (X.t * D.t)) { struct m } : Prop :=
+Fixpoint eq_list (m m' : list (X.t * D.t)) { struct m } : Prop :=
match m, m' with
| nil, nil => True
- | (x,e):l, (x',e')::l' =>
+ | (x,e)::l, (x',e')::l' =>
match X.compare x x' with
| Eq _ => D.eq e e' /\ eq_list l l'
| _ => False
@@ -1112,14 +1112,14 @@ Fixpoint eq_list (m m' list (X.t * D.t)) { struct m } : Prop :=
| _, _ => False
end.
-Definition eq m m' = eq_list m.(this) m'.(this).
+Definition eq m m' := eq_list m.(this) m'.(this).
-Fixpoint lt_list (m m' list (X.t * D.t)) {struct m} : Prop :=
+Fixpoint lt_list (m m' : list (X.t * D.t)) {struct m} : Prop :=
match m, m' with
| nil, nil => False
| nil, _ => True
| _, nil => False
- | (x,e):l, (x',e')::l' =>
+ | (x,e)::l, (x',e')::l' =>
match X.compare x x' with
| Lt _ => True
| Gt _ => False
@@ -1127,9 +1127,9 @@ Fixpoint lt_list (m m' list (X.t * D.t)) {struct m} : Prop :=
end
end.
-Definition lt m m' = lt_list m.(this) m'.(this).
+Definition lt m m' := lt_list m.(this) m'.(this).
-Lemma eq_equal forall m m', eq m m' <-> equal cmp m m' = true.
+Lemma eq_equal : forall m m', eq m m' <-> equal cmp m m' = true.
Proof.
intros (l,Hl); induction l.
intros (l',Hl'); unfold eq; simpl.
@@ -1157,7 +1157,7 @@ Proof.
unfold equal, eq in H6; simpl in H6; auto.
Qed.
-Lemma eq_1 forall m m', Equal cmp m m' -> eq m m'.
+Lemma eq_1 : forall m m', Equal cmp m m' -> eq m m'.
Proof.
intros.
generalize (@equal_1 D.t m m' cmp).
@@ -1165,7 +1165,7 @@ Proof.
intuition.
Qed.
-Lemma eq_2 forall m m', eq m m' -> Equal cmp m m'.
+Lemma eq_2 : forall m m', eq m m' -> Equal cmp m m'.
Proof.
intros.
generalize (@equal_2 D.t m m' cmp).
@@ -1173,7 +1173,7 @@ Proof.
intuition.
Qed.
-Lemma eq_refl forall m : t, eq m m.
+Lemma eq_refl : forall m : t, eq m m.
Proof.
intros (m,Hm); induction m; unfold eq; simpl; auto.
destruct a.
@@ -1186,7 +1186,7 @@ Proof.
apply (MapS.Raw.MX.lt_antirefl l); auto.
Qed.
-Lemma eq_sym forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
+Lemma eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
Proof.
intros (m,Hm); induction m;
intros (m', Hm'); destruct m'; unfold eq; simpl;
@@ -1196,7 +1196,7 @@ Proof.
apply (IHm H0 (Build_slist H4)); auto.
Qed.
-Lemma eq_trans forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
+Lemma eq_trans : forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
Proof.
intros (m1,Hm1); induction m1;
intros (m2, Hm2); destruct m2;
@@ -1213,7 +1213,7 @@ Proof.
apply (IHm1 H1 (Build_slist H6) (Build_slist H8)); intuition.
Qed.
-Lemma lt_trans forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
+Lemma lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
Proof.
intros (m1,Hm1); induction m1;
intros (m2, Hm2); destruct m2;
@@ -1235,7 +1235,7 @@ Proof.
apply (IHm1 H2 (Build_slist H6) (Build_slist H8)); intuition.
Qed.
-Lemma lt_not_eq forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
+Lemma lt_not_eq : forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
Proof.
intros (m1,Hm1); induction m1;
intros (m2, Hm2); destruct m2; unfold eq, lt; simpl;
@@ -1248,9 +1248,9 @@ Proof.
apply (IHm1 H0 (Build_slist H5)); intuition.
Qed.
-Ltac cmp_solve = unfold eq, lt; simpl; try Raw.MX.elim_comp; auto.
+Ltac cmp_solve := unfold eq, lt; simpl; try Raw.MX.elim_comp; auto.
-Definition compare forall m1 m2, Compare lt eq m1 m2.
+Definition compare : forall m1 m2, Compare lt eq m1 m2.
Proof.
intros (m1,Hm1); induction m1;
intros (m2, Hm2); destruct m2;
@@ -1260,9 +1260,9 @@ Proof.
[ apply Lt | | apply Gt ]; cmp_solve.
destruct (D.compare e e');
[ apply Lt | | apply Gt ]; cmp_solve.
- assert (Hm11 sort (Raw.PX.ltk (elt:=D.t)) m1).
+ assert (Hm11 : sort (Raw.PX.ltk (elt:=D.t)) m1).
inversion_clear Hm1; auto.
- assert (Hm22 sort (Raw.PX.ltk (elt:=D.t)) m2).
+ assert (Hm22 : sort (Raw.PX.ltk (elt:=D.t)) m2).
inversion_clear Hm2; auto.
destruct (IHm1 Hm11 (Build_slist Hm22));
[ apply Lt | apply Eq | apply Gt ]; cmp_solve.
diff --git a/theories/FSets/FMapWeak.v b/theories/FSets/FMapWeak.v
index 7baeb2970..9e377d1d0 100644
--- a/theories/FSets/FMapWeak.v
+++ b/theories/FSets/FMapWeak.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSet.v,v 1.2 2004/12/08 19:19:24 letouzey Exp $ *)
Require Export FMapWeakInterface.
Require Export FMapWeakList.
diff --git a/theories/FSets/FMapWeakInterface.v b/theories/FSets/FMapWeakInterface.v
index f5eefec70..22ff07c1f 100644
--- a/theories/FSets/FMapWeakInterface.v
+++ b/theories/FSets/FMapWeakInterface.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FMapWeakInterface.v,v 1.4 2006/02/27 15:39:43 letouzey Exp $ *)
(** * Finite map library *)
@@ -20,81 +20,81 @@ Require Import FSetWeakInterface.
Module Type S.
- Declare Module E DecidableType.
+ Declare Module E : DecidableType.
- Definition key = E.t.
+ Definition key := E.t.
- Parameter t Set -> Set. (** the abstract type of maps *)
+ Parameter t : Set -> Set. (** the abstract type of maps *)
Section Types.
- Variable eltSet.
+ Variable elt:Set.
- Parameter empty t elt.
+ Parameter empty : t elt.
(** The empty map. *)
- Parameter is_empty t elt -> bool.
+ Parameter is_empty : t elt -> bool.
(** Test whether a map is empty or not. *)
- Parameter add key -> elt -> t elt -> t elt.
+ Parameter add : key -> elt -> t elt -> t elt.
(** [add x y m] returns a map containing the same bindings as [m],
plus a binding of [x] to [y]. If [x] was already bound in [m],
its previous binding disappears. *)
- Parameter find key -> t elt -> option elt.
+ Parameter find : key -> t elt -> option elt.
(** [find x m] returns the current binding of [x] in [m],
or raises [Not_found] if no such binding exists.
- NB in Coq, the exception mechanism becomes a option type. *)
+ NB: in Coq, the exception mechanism becomes a option type. *)
- Parameter remove key -> t elt -> t elt.
+ Parameter remove : key -> t elt -> t elt.
(** [remove x m] returns a map containing the same bindings as [m],
except for [x] which is unbound in the returned map. *)
- Parameter mem key -> t elt -> bool.
+ Parameter mem : key -> t elt -> bool.
(** [mem x m] returns [true] if [m] contains a binding for [x],
and [false] otherwise. *)
- (** Coq comment [iter] is useless in a purely functional world *)
- (** val iter (key -> 'a -> unit) -> 'a t -> unit *)
+ (** Coq comment: [iter] is useless in a purely functional world *)
+ (** val iter : (key -> 'a -> unit) -> 'a t -> unit *)
(** iter f m applies f to all bindings in map m. f receives the key as
first argument, and the associated value as second argument.
The bindings are passed to f in increasing order with respect to the
ordering over the type of the keys. Only current bindings are
- presented to f bindings hidden by more recent bindings are not
+ presented to f: bindings hidden by more recent bindings are not
passed to f. *)
- Variable elt' Set.
- Variable elt'' Set.
+ Variable elt' : Set.
+ Variable elt'': Set.
- Parameter map (elt -> elt') -> t elt -> t elt'.
+ Parameter map : (elt -> elt') -> t elt -> t elt'.
(** [map f m] returns a map with same domain as [m], where the associated
value a of all bindings of [m] has been replaced by the result of the
application of [f] to [a]. The bindings are passed to [f] in
increasing order with respect to the ordering over the type of the
keys. *)
- Parameter mapi (key -> elt -> elt') -> t elt -> t elt'.
+ Parameter mapi : (key -> elt -> elt') -> t elt -> t elt'.
(** Same as [S.map], but the function receives as arguments both the
key and the associated value for each binding of the map. *)
- Parameter map2 (option elt -> option elt' -> option elt'') -> t elt -> t elt' -> t elt''.
+ Parameter map2 : (option elt -> option elt' -> option elt'') -> t elt -> t elt' -> t elt''.
(** Not present in Ocaml.
[map f m m'] creates a new map whose bindings belong to the ones of either
[m] or [m']. The presence and value for a key [k] is determined by [f e e']
where [e] and [e'] are the (optional) bindings of [k] in [m] and [m']. *)
- Parameter elements t elt -> list (key*elt).
+ Parameter elements : t elt -> list (key*elt).
(** Not present in Ocaml.
[elements m] returns an assoc list corresponding to the bindings of [m].
Elements of this list are sorted with respect to their first components.
Useful to specify [fold] ... *)
- Parameter fold forall A: Set, (key -> elt -> A -> A) -> t elt -> A -> A.
+ Parameter fold : forall A: Set, (key -> elt -> A -> A) -> t elt -> A -> A.
(** [fold f m a] computes [(f kN dN ... (f k1 d1 a)...)],
where [k1] ... [kN] are the keys of all bindings in [m]
(in increasing order), and [d1] ... [dN] are the associated data. *)
- Parameter equal (elt -> elt -> bool) -> t elt -> t elt -> bool.
+ Parameter equal : (elt -> elt -> bool) -> t elt -> t elt -> bool.
(** [equal cmp m1 m2] tests whether the maps [m1] and [m2] are equal,
that is, contain equal keys and associate them with equal data.
[cmp] is the equality predicate used to compare the data associated
@@ -102,95 +102,95 @@ Module Type S.
Section Spec.
- Variable m m' m'' t elt.
- Variable x y z key.
- Variable e e' elt.
+ Variable m m' m'' : t elt.
+ Variable x y z : key.
+ Variable e e' : elt.
- Parameter MapsTo key -> elt -> t elt -> Prop.
+ Parameter MapsTo : key -> elt -> t elt -> Prop.
- Definition In (kkey)(m: t elt) : Prop := exists e:elt, MapsTo k e m.
+ Definition In (k:key)(m: t elt) : Prop := exists e:elt, MapsTo k e m.
- Definition Empty m = forall (a : key)(e:elt) , ~ MapsTo a e m.
+ Definition Empty m := forall (a : key)(e:elt) , ~ MapsTo a e m.
- Definition eq_key (p p'key*elt) := E.eq (fst p) (fst p').
+ Definition eq_key (p p':key*elt) := E.eq (fst p) (fst p').
- Definition eq_key_elt (p p'key*elt) :=
+ Definition eq_key_elt (p p':key*elt) :=
E.eq (fst p) (fst p') /\ (snd p) = (snd p').
(** Specification of [MapsTo] *)
- Parameter MapsTo_1 E.eq x y -> MapsTo x e m -> MapsTo y e m.
+ Parameter MapsTo_1 : E.eq x y -> MapsTo x e m -> MapsTo y e m.
(** Specification of [mem] *)
- Parameter mem_1 In x m -> mem x m = true.
- Parameter mem_2 mem x m = true -> In x m.
+ Parameter mem_1 : In x m -> mem x m = true.
+ Parameter mem_2 : mem x m = true -> In x m.
(** Specification of [empty] *)
- Parameter empty_1 Empty empty.
+ Parameter empty_1 : Empty empty.
(** Specification of [is_empty] *)
- Parameter is_empty_1 Empty m -> is_empty m = true.
- Parameter is_empty_2 is_empty m = true -> Empty m.
+ Parameter is_empty_1 : Empty m -> is_empty m = true.
+ Parameter is_empty_2 : is_empty m = true -> Empty m.
(** Specification of [add] *)
- Parameter add_1 E.eq x y -> MapsTo y e (add x e m).
- Parameter add_2 ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
- Parameter add_3 ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
+ Parameter add_1 : E.eq x y -> MapsTo y e (add x e m).
+ Parameter add_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
+ Parameter add_3 : ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
(** Specification of [remove] *)
- Parameter remove_1 E.eq x y -> ~ In y (remove x m).
- Parameter remove_2 ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
- Parameter remove_3 MapsTo y e (remove x m) -> MapsTo y e m.
+ Parameter remove_1 : E.eq x y -> ~ In y (remove x m).
+ Parameter remove_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
+ Parameter remove_3 : MapsTo y e (remove x m) -> MapsTo y e m.
(** Specification of [find] *)
- Parameter find_1 MapsTo x e m -> find x m = Some e.
- Parameter find_2 find x m = Some e -> MapsTo x e m.
+ Parameter find_1 : MapsTo x e m -> find x m = Some e.
+ Parameter find_2 : find x m = Some e -> MapsTo x e m.
(** Specification of [elements] *)
- Parameter elements_1
+ Parameter elements_1 :
MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
- Parameter elements_2
+ Parameter elements_2 :
InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
- Parameter elements_3 noredunA eq_key (elements m).
+ Parameter elements_3 : noredunA eq_key (elements m).
(** Specification of [fold] *)
- Parameter fold_1
- forall (A Set) (i : A) (f : key -> elt -> A -> A),
+ Parameter fold_1 :
+ forall (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.
- Definition Equal cmp m m' =
+ Definition Equal cmp 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).
- Variable cmp elt -> elt -> bool.
+ Variable cmp : elt -> elt -> bool.
(** Specification of [equal] *)
- Parameter equal_1 Equal cmp m m' -> equal cmp m m' = true.
- Parameter equal_2 equal cmp m m' = true -> Equal cmp m m'.
+ Parameter equal_1 : Equal cmp m m' -> equal cmp m m' = true.
+ Parameter equal_2 : equal cmp m m' = true -> Equal cmp m m'.
End Spec.
End Types.
(** Specification of [map] *)
- Parameter map_1 forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:elt->elt'),
+ Parameter 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).
- Parameter map_2 forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'),
+ Parameter map_2 : forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'),
In x (map f m) -> In x m.
(** Specification of [mapi] *)
- Parameter mapi_1 forall (elt elt':Set)(m: t elt)(x:key)(e:elt)
- (fkey->elt->elt'), MapsTo x e m ->
+ Parameter 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).
- Parameter mapi_2 forall (elt elt':Set)(m: t elt)(x:key)
- (fkey->elt->elt'), In x (mapi f m) -> In x m.
+ Parameter mapi_2 : forall (elt elt':Set)(m: t elt)(x:key)
+ (f:key->elt->elt'), In x (mapi f m) -> In x m.
(** Specification of [map2] *)
- Parameter map2_1 forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
- (xkey)(f:option elt->option elt'->option elt''),
+ Parameter 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').
- Parameter map2_2 forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
- (xkey)(f:option elt->option elt'->option elt''),
+ Parameter 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'.
Hint Immediate MapsTo_1 mem_2 is_empty_2.
diff --git a/theories/FSets/FMapWeakList.v b/theories/FSets/FMapWeakList.v
index ff54a928a..6c544053e 100644
--- a/theories/FSets/FMapWeakList.v
+++ b/theories/FSets/FMapWeakList.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSet.v,v 1.2 2004/12/08 19:19:24 letouzey Exp $ *)
(** * Finite map library *)
@@ -22,37 +22,37 @@ Unset Strict Implicit.
Arguments Scope list [type_scope].
-Module Raw (XDecidableType).
+Module Raw (X:DecidableType).
-Module PX = PairDecidableType X.
+Module PX := PairDecidableType X.
Import PX.
-Definition key = X.t.
-Definition t (eltSet) := list (X.t * elt).
+Definition key := X.t.
+Definition t (elt:Set) := list (X.t * elt).
Section Elt.
-Variable elt Set.
+Variable elt : Set.
-(* now in PairDecidableType
-Definition eqk (p p'key*elt) := X.eq (fst p) (fst p').
-Definition eqke (p p'key*elt) :=
+(* now in PairDecidableType:
+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').
*)
-Notation eqk = (eqk (elt:=elt)).
-Notation eqke = (eqke (elt:=elt)).
-Notation MapsTo = (MapsTo (elt:=elt)).
-Notation In = (In (elt:=elt)).
-Notation noredunA = (noredunA eqk).
+Notation eqk := (eqk (elt:=elt)).
+Notation eqke := (eqke (elt:=elt)).
+Notation MapsTo := (MapsTo (elt:=elt)).
+Notation In := (In (elt:=elt)).
+Notation noredunA := (noredunA eqk).
(** * [empty] *)
-Definition empty t elt := nil.
+Definition empty : t elt := nil.
-Definition Empty m = forall (a : key)(e:elt), ~ MapsTo a e m.
+Definition Empty m := forall (a : key)(e:elt), ~ MapsTo a e m.
-Lemma empty_1 Empty empty.
+Lemma empty_1 : Empty empty.
Proof.
unfold Empty,empty.
intros a e.
@@ -62,26 +62,26 @@ Qed.
Hint Resolve empty_1.
-Lemma empty_noredun noredunA empty.
+Lemma empty_noredun : noredunA empty.
Proof.
unfold empty; auto.
Qed.
(** * [is_empty] *)
-Definition is_empty (l t elt) : bool := if l then true else false.
+Definition is_empty (l : t elt) : bool := if l then true else false.
-Lemma is_empty_1 forall m, Empty m -> is_empty m = true.
+Lemma is_empty_1 :forall m, Empty m -> is_empty m = true.
Proof.
unfold Empty, PX.MapsTo.
intros m.
case m;auto.
intros p l inlist.
destruct p.
- absurd (InA eqke (t0, e) ((t0, e) : l));auto.
+ absurd (InA eqke (t0, e) ((t0, e) :: l));auto.
Qed.
-Lemma is_empty_2 forall m, is_empty m = true -> Empty m.
+Lemma is_empty_2 : forall m, is_empty m = true -> Empty m.
Proof.
intros m.
case m;auto.
@@ -91,13 +91,13 @@ Qed.
(** * [mem] *)
-Fixpoint mem (k key) (s : t elt) {struct s} : bool :=
+Fixpoint 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
+ | (k',_) :: l => if X.eq_dec k k' then true else mem k l
end.
-Lemma mem_1 forall m (Hm:noredunA m) x, In x m -> mem x m = true.
+Lemma mem_1 : forall m (Hm:noredunA m) x, In x m -> mem x m = true.
Proof.
intros m Hm x; generalize Hm; clear Hm.
functional induction mem x m;intros noredun belong1;trivial.
@@ -111,7 +111,7 @@ Proof.
exists x; auto.
Qed.
-Lemma mem_2 forall m (Hm:noredunA m) x, mem x m = true -> In x m.
+Lemma mem_2 : forall m (Hm:noredunA 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 noredun hyp; try discriminate.
@@ -123,19 +123,19 @@ Qed.
(** * [find] *)
-Fixpoint find (kkey) (s: t elt) {struct s} : option elt :=
+Fixpoint 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'
+ | (k',x)::s' => if X.eq_dec k k' then Some x else find k s'
end.
-Lemma find_2 forall m x e, find x m = Some e -> MapsTo x e m.
+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.
Qed.
-Lemma find_1 forall m (Hm:noredunA m) x e,
+Lemma find_1 : forall m (Hm:noredunA 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.
@@ -153,7 +153,7 @@ Qed.
(* Not part of the exported specifications, used later for [combine]. *)
-Lemma find_eq forall m (Hm:noredunA m) x x',
+Lemma find_eq : forall m (Hm:noredunA m) x x',
X.eq x x' -> find x m = find x' m.
Proof.
induction m; simpl; auto; destruct a; intros.
@@ -166,19 +166,19 @@ Qed.
(** * [add] *)
-Fixpoint add (k key) (x : elt) (s : t elt) {struct s} : t elt :=
+Fixpoint 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
+ | nil => (k,x) :: nil
+ | (k',y) :: l => if X.eq_dec k k' then (k,x)::l else (k',y)::add k x l
end.
-Lemma add_1 forall m x y e, X.eq x y -> MapsTo y e (add x e m).
+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.
Qed.
-Lemma add_2 forall m x y e e',
+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.
@@ -190,7 +190,7 @@ Proof.
intros y' e' eqky'; inversion_clear 1; intuition.
Qed.
-Lemma add_3 forall m x y e e',
+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.
@@ -200,7 +200,7 @@ Proof.
inversion_clear 2; auto.
Qed.
-Lemma add_3' forall m x y e e',
+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.
@@ -213,7 +213,7 @@ Proof.
inversion_clear 2; auto.
Qed.
-Lemma add_noredun forall m (Hm:noredunA m) x e, noredunA (add x e m).
+Lemma add_noredun : forall m (Hm:noredunA m) x e, noredunA (add x e m).
Proof.
induction m.
simpl; constructor; auto; red; inversion 1.
@@ -229,7 +229,7 @@ Qed.
(* Not part of the exported specifications, used later for [combine]. *)
-Lemma add_eq forall m (Hm:noredunA m) x a e,
+Lemma add_eq : forall m (Hm:noredunA m) x a e,
X.eq x a -> find x (add a e m) = Some e.
Proof.
intros.
@@ -238,7 +238,7 @@ Proof.
apply add_1; auto.
Qed.
-Lemma add_not_eq forall m (Hm:noredunA m) x a e,
+Lemma add_not_eq : forall m (Hm:noredunA m) x a e,
~X.eq x a -> find x (add a e m) = find x m.
Proof.
intros.
@@ -257,13 +257,13 @@ Qed.
(** * [remove] *)
-Fixpoint remove (k key) (s : t elt) {struct s} : t elt :=
+Fixpoint 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
+ | (k',x) :: l => if X.eq_dec k k' then l else (k',x) :: remove k l
end.
-Lemma remove_1 forall m (Hm:noredunA m) x y, X.eq x y -> ~ In y (remove x m).
+Lemma remove_1 : forall m (Hm:noredunA 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.
@@ -286,7 +286,7 @@ Proof.
exists e; auto.
Qed.
-Lemma remove_2 forall m (Hm:noredunA m) x y e,
+Lemma remove_2 : forall m (Hm:noredunA 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.
@@ -298,7 +298,7 @@ Proof.
inversion_clear 1; inversion_clear 2; auto.
Qed.
-Lemma remove_3 forall m (Hm:noredunA m) x y e,
+Lemma remove_3 : forall m (Hm:noredunA 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.
@@ -306,7 +306,7 @@ Proof.
do 2 inversion_clear 1; auto.
Qed.
-Lemma remove_3' forall m (Hm:noredunA m) x y e,
+Lemma remove_3' : forall m (Hm:noredunA 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.
@@ -314,7 +314,7 @@ Proof.
do 2 inversion_clear 1; auto.
Qed.
-Lemma remove_noredun forall m (Hm:noredunA m) x, noredunA (remove x m).
+Lemma remove_noredun : forall m (Hm:noredunA m) x, noredunA (remove x m).
Proof.
induction m.
simpl; intuition.
@@ -328,33 +328,33 @@ Qed.
(** * [elements] *)
-Definition elements (m t elt) := m.
+Definition elements (m: t elt) := m.
-Lemma elements_1 forall m x e, MapsTo x e m -> InA eqke (x,e) (elements m).
+Lemma elements_1 : forall m x e, MapsTo x e m -> InA eqke (x,e) (elements m).
Proof.
auto.
Qed.
-Lemma elements_2 forall m x e, InA eqke (x,e) (elements m) -> MapsTo x e m.
+Lemma elements_2 : forall m x e, InA eqke (x,e) (elements m) -> MapsTo x e m.
Proof.
auto.
Qed.
-Lemma elements_3 forall m (Hm:noredunA m), noredunA (elements m).
+Lemma elements_3 : forall m (Hm:noredunA m), noredunA (elements m).
Proof.
auto.
Qed.
(** * [fold] *)
-Fixpoint fold (ASet)(f:key->elt->A->A)(m:t elt) {struct m} : A -> A :=
+Fixpoint fold (A:Set)(f:key->elt->A->A)(m:t elt) {struct m} : A -> A :=
fun acc =>
match m with
| nil => acc
- | (k,e):m' => fold f m' (f k e acc)
+ | (k,e)::m' => fold f m' (f k e acc)
end.
-Lemma fold_1 forall m (A:Set)(i:A)(f:key->elt->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.
@@ -362,27 +362,27 @@ Qed.
(** * [equal] *)
-Definition check (cmp elt -> elt -> bool)(k:key)(e:elt)(m': t elt) :=
+Definition check (cmp : elt -> elt -> bool)(k:key)(e:elt)(m': t elt) :=
match find k m' with
| None => false
| Some e' => cmp e e'
end.
-Definition submap (cmp elt -> elt -> bool)(m m' : t elt) : bool :=
+Definition submap (cmp : elt -> elt -> bool)(m m' : t elt) : bool :=
fold (fun k e b => andb (check cmp k e m') b) m true.
-Definition equal (cmp elt -> elt -> bool)(m m' : t elt) : bool :=
+Definition equal (cmp : elt -> elt -> bool)(m m' : t elt) : bool :=
andb (submap cmp m m') (submap (fun e' e => cmp e e') m' m).
-Definition Submap cmp m m' =
+Definition Submap cmp 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).
-Definition Equal cmp m m' =
+Definition Equal cmp 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).
-Lemma submap_1 forall m (Hm:noredunA m) m' (Hm': noredunA m') cmp,
+Lemma submap_1 : forall m (Hm:noredunA m) m' (Hm': noredunA m') cmp,
Submap cmp m m' -> submap cmp m m' = true.
Proof.
unfold Submap, submap.
@@ -391,7 +391,7 @@ Proof.
destruct a; simpl; intros.
destruct H.
inversion_clear Hm.
- assert (H3 In t0 m').
+ assert (H3 : In t0 m').
apply H; exists e; auto.
destruct H3 as (e', H3).
unfold check at 2; rewrite (find_1 Hm' H3).
@@ -403,7 +403,7 @@ Proof.
apply H0 with k; auto.
Qed.
-Lemma submap_2 forall m (Hm:noredunA m) m' (Hm': noredunA m') cmp,
+Lemma submap_2 : forall m (Hm:noredunA m) m' (Hm': noredunA m') cmp,
submap cmp m m' = true -> Submap cmp m m'.
Proof.
unfold Submap, submap.
@@ -418,7 +418,7 @@ Proof.
rewrite andb_b_true in H.
assert (check cmp t0 e m' = true).
clear H1 H0 Hm' IHm.
- set (b=check cmp t0 e m') in *.
+ set (b:=check cmp t0 e m') in *.
generalize H; clear H; generalize b; clear b.
induction m; simpl; auto; intros.
destruct a; simpl in *.
@@ -444,7 +444,7 @@ Qed.
(** Specification of [equal] *)
-Lemma equal_1 forall m (Hm:noredunA m) m' (Hm': noredunA m') cmp,
+Lemma equal_1 : forall m (Hm:noredunA m) m' (Hm': noredunA m') cmp,
Equal cmp m m' -> equal cmp m m' = true.
Proof.
unfold Equal, equal.
@@ -452,7 +452,7 @@ Proof.
apply andb_true_intro; split; apply submap_1; unfold Submap; firstorder.
Qed.
-Lemma equal_2 forall m (Hm:noredunA m) m' (Hm':noredunA m') cmp,
+Lemma equal_2 : forall m (Hm:noredunA m) m' (Hm':noredunA m') cmp,
equal cmp m m' = true -> Equal cmp m m'.
Proof.
unfold Equal, equal.
@@ -463,20 +463,20 @@ Proof.
firstorder.
Qed.
-Variable elt'Set.
+Variable elt':Set.
(** * [map] and [mapi] *)
-Fixpoint map (felt -> elt') (m:t elt) {struct m} : t elt' :=
+Fixpoint map (f:elt -> elt') (m:t elt) {struct m} : t elt' :=
match m with
| nil => nil
- | (k,e):m' => (k,f e) :: map f m'
+ | (k,e)::m' => (k,f e) :: map f m'
end.
-Fixpoint mapi (f key -> elt -> elt') (m:t elt) {struct m} : t elt' :=
+Fixpoint mapi (f: key -> elt -> elt') (m:t elt) {struct m} : t elt' :=
match m with
| nil => nil
- | (k,e):m' => (k,f k e) :: mapi f m'
+ | (k,e)::m' => (k,f k e) :: mapi f m'
end.
End Elt.
@@ -484,11 +484,11 @@ Section Elt2.
(* A new section is necessary for previous definitions to work
with different [elt], especially [MapsTo]... *)
-Variable elt elt' Set.
+Variable elt elt' : Set.
(** Specification of [map] *)
-Lemma map_1 forall (m:t elt)(x:key)(e:elt)(f:elt->elt'),
+Lemma map_1 : forall (m:t elt)(x:key)(e:elt)(f:elt->elt'),
MapsTo x e m -> MapsTo x (f e) (map f m).
Proof.
intros m x e f.
@@ -505,7 +505,7 @@ Proof.
unfold MapsTo in *; auto.
Qed.
-Lemma map_2 forall (m:t elt)(x:key)(f:elt->elt'),
+Lemma map_2 : forall (m:t elt)(x:key)(f:elt->elt'),
In x (map f m) -> In x m.
Proof.
intros m x f.
@@ -526,7 +526,7 @@ Proof.
constructor 2; auto.
Qed.
-Lemma map_noredun forall m (Hm : noredunA (@eqk elt) m)(f:elt->elt'),
+Lemma map_noredun : forall m (Hm : noredunA (@eqk elt) m)(f:elt->elt'),
noredunA (@eqk elt') (map f m).
Proof.
induction m; simpl; auto.
@@ -544,7 +544,7 @@ Qed.
(** Specification of [mapi] *)
-Lemma mapi_1 forall (m:t elt)(x:key)(e:elt)(f:key->elt->elt'),
+Lemma mapi_1 : forall (m:t elt)(x:key)(e:elt)(f:key->elt->elt'),
MapsTo x e m ->
exists y, X.eq y x /\ MapsTo x (f y e) (mapi f m).
Proof.
@@ -565,7 +565,7 @@ Proof.
exists y; intuition.
Qed.
-Lemma mapi_2 forall (m:t elt)(x:key)(f:key->elt->elt'),
+Lemma mapi_2 : forall (m:t elt)(x:key)(f:key->elt->elt'),
In x (mapi f m) -> In x m.
Proof.
intros m x f.
@@ -586,7 +586,7 @@ Proof.
constructor 2; auto.
Qed.
-Lemma mapi_noredun forall m (Hm : noredunA (@eqk elt) m)(f: key->elt->elt'),
+Lemma mapi_noredun : forall m (Hm : noredunA (@eqk elt) m)(f: key->elt->elt'),
noredunA (@eqk elt') (mapi f m).
Proof.
induction m; simpl; auto.
@@ -604,28 +604,28 @@ Qed.
End Elt2.
Section Elt3.
-Variable elt elt' elt'' Set.
+Variable elt elt' elt'' : Set.
-Notation oee' = (option elt * option elt')%type.
+Notation oee' := (option elt * option elt')%type.
-Definition combine_l (mt elt)(m':t elt') : t oee' :=
+Definition combine_l (m:t elt)(m':t elt') : t oee' :=
mapi (fun k e => (Some e, find k m')) m.
-Definition combine_r (mt elt)(m':t elt') : t oee' :=
+Definition combine_r (m:t elt)(m':t elt') : t oee' :=
mapi (fun k e' => (find k m, Some e')) m'.
-Definition fold_right_pair (A B CSet)(f:A->B->C->C)(l:list (A*B))(i:C) :=
+Definition fold_right_pair (A B C:Set)(f:A->B->C->C)(l:list (A*B))(i:C) :=
List.fold_right (fun p => f (fst p) (snd p)) i l.
-Definition combine (mt elt)(m':t elt') : t oee' :=
- let l = combine_l m m' in
- let r = combine_r m m' in
- fold_right_pair (add (elt=oee')) l r.
+Definition combine (m:t elt)(m':t elt') : t oee' :=
+ let l := combine_l m m' in
+ let r := combine_r m m' in
+ fold_right_pair (add (elt:=oee')) l r.
-Lemma fold_right_pair_noredun
- forall l r (Hl noredunA (eqk (elt:=oee')) l)
- (Hl noredunA (eqk (elt:=oee')) r),
- noredunA (eqk (elt=oee')) (fold_right_pair (add (elt:=oee')) l r).
+Lemma fold_right_pair_noredun :
+ forall l r (Hl: noredunA (eqk (elt:=oee')) l)
+ (Hl: noredunA (eqk (elt:=oee')) r),
+ noredunA (eqk (elt:=oee')) (fold_right_pair (add (elt:=oee')) l r).
Proof.
induction l; simpl; auto.
destruct a; simpl; auto.
@@ -634,35 +634,35 @@ Proof.
Qed.
Hint Resolve fold_right_pair_noredun.
-Lemma combine_noredun
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m'),
+Lemma combine_noredun :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m'),
noredunA (@eqk oee') (combine m m').
Proof.
unfold combine, combine_r, combine_l.
intros.
- set (f1 = fun (k : key) (e : elt) => (Some e, find k m')).
- set (f2 = fun (k : key) (e' : elt') => (find k m, Some e')).
+ set (f1 := fun (k : key) (e : elt) => (Some e, find k m')).
+ set (f2 := fun (k : key) (e' : elt') => (find k m, Some e')).
generalize (mapi_noredun Hm f1).
generalize (mapi_noredun Hm' f2).
- set (l = mapi f1 m); clearbody l.
- set (r = mapi f2 m'); clearbody r.
+ set (l := mapi f1 m); clearbody l.
+ set (r := mapi f2 m'); clearbody r.
auto.
Qed.
-Definition at_least_left (ooption elt)(o':option elt') :=
+Definition at_least_left (o:option elt)(o':option elt') :=
match o with
| None => None
| _ => Some (o,o')
end.
-Definition at_least_right (ooption elt)(o':option elt') :=
+Definition at_least_right (o:option elt)(o':option elt') :=
match o' with
| None => None
| _ => Some (o,o')
end.
-Lemma combine_l_1
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
+Lemma combine_l_1 :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
find x (combine_l m m') = at_least_left (find x m) (find x m').
Proof.
unfold combine_l.
@@ -680,8 +680,8 @@ Proof.
rewrite (find_1 Hm H1) in H; discriminate.
Qed.
-Lemma combine_r_1
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
+Lemma combine_r_1 :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
find x (combine_r m m') = at_least_right (find x m) (find x m').
Proof.
unfold combine_r.
@@ -699,28 +699,28 @@ Proof.
rewrite (find_1 Hm' H1) in H; discriminate.
Qed.
-Definition at_least_one (ooption elt)(o':option elt') :=
+Definition at_least_one (o:option elt)(o':option elt') :=
match o, o' with
| None, None => None
| _, _ => Some (o,o')
end.
-Lemma combine_1
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
+Lemma combine_1 :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
find x (combine m m') = at_least_one (find x m) (find x m').
Proof.
unfold combine.
intros.
generalize (combine_r_1 Hm Hm' x).
generalize (combine_l_1 Hm Hm' x).
- assert (noredunA (eqk (elt=oee')) (combine_l m m')).
+ assert (noredunA (eqk (elt:=oee')) (combine_l m m')).
unfold combine_l; apply mapi_noredun; auto.
- assert (noredunA (eqk (elt=oee')) (combine_r m m')).
+ assert (noredunA (eqk (elt:=oee')) (combine_r m m')).
unfold combine_r; apply mapi_noredun; auto.
- set (l = combine_l m m') in *; clearbody l.
- set (r = combine_r m m') in *; clearbody r.
- set (o = find x m); clearbody o.
- set (o' = find x m'); clearbody o'.
+ set (l := combine_l m m') in *; clearbody l.
+ set (r := combine_r m m') in *; clearbody r.
+ set (o := find x m); clearbody o.
+ set (o' := find x m'); clearbody o'.
clear Hm' Hm m m'.
induction l.
destruct o; destruct o'; simpl; intros; discriminate || auto.
@@ -736,30 +736,30 @@ Proof.
apply add_not_eq; auto.
Qed.
-Variable f option elt -> option elt' -> option elt''.
+Variable f : option elt -> option elt' -> option elt''.
-Definition option_cons (ASet)(k:key)(o:option A)(l:list (key*A)) :=
+Definition option_cons (A:Set)(k:key)(o:option A)(l:list (key*A)) :=
match o with
- | Some e => (k,e):l
+ | Some e => (k,e)::l
| None => l
end.
-Definition map2 m m' =
- let m0 t oee' := combine m m' in
- let m1 t (option elt'') := map (fun p => f (fst p) (snd p)) m0 in
- fold_right_pair (option_cons (A=elt'')) m1 nil.
+Definition map2 m m' :=
+ let m0 : t oee' := combine m m' in
+ let m1 : t (option elt'') := map (fun p => f (fst p) (snd p)) m0 in
+ fold_right_pair (option_cons (A:=elt'')) m1 nil.
-Lemma map2_noredun
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m'),
+Lemma map2_noredun :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m'),
noredunA (@eqk elt'') (map2 m m').
Proof.
intros.
unfold map2.
- assert (H0=combine_noredun Hm Hm').
- set (l0=combine m m') in *; clearbody l0.
- set (f'= fun p : oee' => f (fst p) (snd p)).
- assert (H1=map_noredun (elt' := option elt'') H0 f').
- set (l1=map f' l0) in *; clearbody l1.
+ assert (H0:=combine_noredun Hm Hm').
+ set (l0:=combine m m') in *; clearbody l0.
+ set (f':= fun p : oee' => f (fst p) (snd p)).
+ assert (H1:=map_noredun (elt' := option elt'') H0 f').
+ set (l1:=map f' l0) in *; clearbody l1.
clear f' f H0 l0 Hm Hm' m m'.
induction l1.
simpl; auto.
@@ -775,24 +775,24 @@ Proof.
inversion_clear H1; auto.
Qed.
-Definition at_least_one_then_f (ooption elt)(o':option elt') :=
+Definition at_least_one_then_f (o:option elt)(o':option elt') :=
match o, o' with
| None, None => None
| _, _ => f o o'
end.
-Lemma map2_0
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
+Lemma map2_0 :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
find x (map2 m m') = at_least_one_then_f (find x m) (find x m').
Proof.
intros.
unfold map2.
- assert (H=combine_1 Hm Hm' x).
- assert (H2=combine_noredun Hm Hm').
- set (f'= fun p : oee' => f (fst p) (snd p)).
- set (m0 = combine m m') in *; clearbody m0.
- set (o=find x m) in *; clearbody o.
- set (o'=find x m') in *; clearbody o'.
+ assert (H:=combine_1 Hm Hm' x).
+ assert (H2:=combine_noredun Hm Hm').
+ set (f':= fun p : oee' => f (fst p) (snd p)).
+ set (m0 := combine m m') in *; clearbody m0.
+ set (o:=find x m) in *; clearbody o.
+ set (o':=find x m') in *; clearbody o'.
clear Hm Hm' m m'.
generalize H; clear H.
match goal with |- ?m=?n -> ?p=?q =>
@@ -838,8 +838,8 @@ Proof.
Qed.
(** Specification of [map2] *)
-Lemma map2_1
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
+Lemma map2_1 :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
In x m \/ In x m' ->
find x (map2 m m') = f (find x m) (find x m').
Proof.
@@ -852,8 +852,8 @@ Proof.
destruct (find x m); simpl; auto.
Qed.
-Lemma map2_2
- forall m (HmnoredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
+Lemma map2_2 :
+ forall m (Hm:noredunA (@eqk elt) m) m' (Hm':noredunA (@eqk elt') m')(x:key),
In x (map2 m m') -> In x m \/ In x m'.
Proof.
intros.
@@ -874,85 +874,85 @@ End Elt3.
End Raw.
-Module Make (X DecidableType) <: S with Module E:=X.
- Module Raw = Raw X.
+Module Make (X: DecidableType) <: S with Module E:=X.
+ Module Raw := Raw X.
- Module E = X.
- Definition key = X.t.
+ Module E := X.
+ Definition key := X.t.
- Record slist (eltSet) : Set :=
- {this > Raw.t elt; noredun : noredunA (@Raw.PX.eqk elt) this}.
- Definition t (eltSet) := slist elt.
+ Record slist (elt:Set) : Set :=
+ {this :> Raw.t elt; noredun : noredunA (@Raw.PX.eqk elt) this}.
+ Definition t (elt:Set) := slist elt.
Section Elt.
- Variable elt elt' elt''Set.
-
- Implicit Types m t elt.
-
- Definition empty = Build_slist (Raw.empty_noredun elt).
- Definition is_empty m = Raw.is_empty m.(this).
- Definition add x e m = Build_slist (Raw.add_noredun m.(noredun) x e).
- Definition find x m = Raw.find x m.(this).
- Definition remove x m = Build_slist (Raw.remove_noredun m.(noredun) x).
- Definition mem x m = Raw.mem x m.(this).
- Definition map f m t elt' := Build_slist (Raw.map_noredun m.(noredun) f).
- Definition mapi f m t elt' := Build_slist (Raw.mapi_noredun m.(noredun) f).
- Definition map2 f m (m't elt') : t elt'' :=
+ Variable elt elt' elt'':Set.
+
+ Implicit Types m : t elt.
+
+ Definition empty := Build_slist (Raw.empty_noredun elt).
+ Definition is_empty m := Raw.is_empty m.(this).
+ Definition add x e m := Build_slist (Raw.add_noredun m.(noredun) x e).
+ Definition find x m := Raw.find x m.(this).
+ Definition remove x m := Build_slist (Raw.remove_noredun m.(noredun) x).
+ Definition mem x m := Raw.mem x m.(this).
+ Definition map f m : t elt' := Build_slist (Raw.map_noredun m.(noredun) f).
+ Definition mapi f m : t elt' := Build_slist (Raw.mapi_noredun m.(noredun) f).
+ Definition map2 f m (m':t elt') : t elt'' :=
Build_slist (Raw.map2_noredun f m.(noredun) m'.(noredun)).
- 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 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).
+ 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).
- Definition eq_key (p p'key*elt) := X.eq (fst p) (fst p').
+ Definition eq_key (p p':key*elt) := X.eq (fst p) (fst p').
- Definition eq_key_elt (p p'key*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 MapsTo_1 m := @Raw.PX.MapsTo_eq elt m.(this).
- Definition mem_1 m = @Raw.mem_1 elt m.(this) m.(noredun).
- Definition mem_2 m = @Raw.mem_2 elt m.(this) m.(noredun).
+ Definition mem_1 m := @Raw.mem_1 elt m.(this) m.(noredun).
+ Definition mem_2 m := @Raw.mem_2 elt m.(this) m.(noredun).
- Definition empty_1 = @Raw.empty_1.
+ 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 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).
+ 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).
- Definition remove_1 m = @Raw.remove_1 elt m.(this) m.(noredun).
- Definition remove_2 m = @Raw.remove_2 elt m.(this) m.(noredun).
- Definition remove_3 m = @Raw.remove_3 elt m.(this) m.(noredun).
+ Definition remove_1 m := @Raw.remove_1 elt m.(this) m.(noredun).
+ Definition remove_2 m := @Raw.remove_2 elt m.(this) m.(noredun).
+ Definition remove_3 m := @Raw.remove_3 elt m.(this) m.(noredun).
- Definition find_1 m = @Raw.find_1 elt m.(this) m.(noredun).
- Definition find_2 m = @Raw.find_2 elt m.(this).
+ Definition find_1 m := @Raw.find_1 elt m.(this) m.(noredun).
+ 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.(noredun).
+ 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.(noredun).
- Definition fold_1 m = @Raw.fold_1 elt m.(this).
+ 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 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 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 :=
+ Definition map2_1 m (m':t elt') x f :=
@Raw.map2_1 elt elt' elt'' f m.(this) m.(noredun) m'.(this) m'.(noredun) x.
- Definition map2_2 m (m't elt') x f :=
+ Definition map2_2 m (m':t elt') x f :=
@Raw.map2_2 elt elt' elt'' f m.(this) m.(noredun) m'.(this) m'.(noredun) x.
- Definition equal_1 m m' = @Raw.equal_1 elt m.(this) m.(noredun) m'.(this) m'.(noredun).
- Definition equal_2 m m' = @Raw.equal_2 elt m.(this) m.(noredun) m'.(this) m'.(noredun).
+ Definition equal_1 m m' := @Raw.equal_1 elt m.(this) m.(noredun) m'.(this) m'.(noredun).
+ Definition equal_2 m m' := @Raw.equal_2 elt m.(this) m.(noredun) m'.(this) m'.(noredun).
End Elt.
End Make.
diff --git a/theories/FSets/FSet.v b/theories/FSets/FSet.v
index f689462ba..232eb2912 100644
--- a/theories/FSets/FSet.v
+++ b/theories/FSets/FSet.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSet.v,v 1.2 2004/12/08 19:19:24 letouzey Exp $ *)
Require Export OrderedType.
Require Export FSetInterface.
@@ -14,4 +14,3 @@ Require Export FSetBridge.
Require Export FSetProperties.
Require Export FSetEqProperties.
Require Export FSetList.
-
diff --git a/theories/FSets/FSetBridge.v b/theories/FSets/FSetBridge.v
index 05457b5a5..4296f619f 100644
--- a/theories/FSets/FSetBridge.v
+++ b/theories/FSets/FSetBridge.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetBridge.v,v 1.6 2006/03/09 18:34:51 letouzey Exp $ *)
(** * Finite sets library *)
@@ -20,33 +20,33 @@ Set Firstorder Depth 2.
(** * From non-dependent signature [S] to dependent signature [Sdep]. *)
-Module DepOfNodep (M S) <: Sdep with Module E := M.E.
+Module DepOfNodep (M: S) <: Sdep with Module E := M.E.
Import M.
- Module ME = OrderedTypeFacts E.
+ Module ME := OrderedTypeFacts E.
- Definition empty {s : t | Empty s}.
+ Definition empty : {s : t | Empty s}.
Proof.
exists empty; auto.
Qed.
- Definition is_empty forall s : t, {Empty s} + {~ Empty s}.
+ Definition is_empty : forall s : t, {Empty s} + {~ Empty s}.
Proof.
- intros; generalize (is_empty_1 (s=s)) (is_empty_2 (s:=s)).
+ intros; generalize (is_empty_1 (s:=s)) (is_empty_2 (s:=s)).
case (is_empty s); intuition.
Qed.
- Definition mem forall (x : elt) (s : t), {In x s} + {~ In x s}.
+ Definition mem : forall (x : elt) (s : t), {In x s} + {~ In x s}.
Proof.
- intros; generalize (mem_1 (s=s) (x:=x)) (mem_2 (s:=s) (x:=x)).
+ intros; generalize (mem_1 (s:=s) (x:=x)) (mem_2 (s:=s) (x:=x)).
case (mem x s); intuition.
Qed.
- Definition Add (x elt) (s s' : t) :=
- forall y elt, In y s' <-> E.eq x y \/ In y s.
+ Definition Add (x : elt) (s s' : t) :=
+ forall y : elt, In y s' <-> E.eq x y \/ In y s.
- Definition add forall (x : elt) (s : t), {s' : t | Add x s s'}.
+ Definition add : forall (x : elt) (s : t), {s' : t | Add x s s'}.
Proof.
intros; exists (add x s); auto.
unfold Add in |- *; intuition.
@@ -55,15 +55,15 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
eapply add_3; eauto.
Qed.
- Definition singleton
- forall x elt, {s : t | forall y : elt, In y s <-> E.eq x y}.
+ Definition singleton :
+ forall x : elt, {s : t | forall y : elt, In y s <-> E.eq x y}.
Proof.
intros; exists (singleton x); intuition.
Qed.
- Definition remove
- forall (x elt) (s : t),
- {s' t | forall y : elt, In y s' <-> ~ E.eq x y /\ In y s}.
+ Definition remove :
+ forall (x : elt) (s : t),
+ {s' : t | forall y : elt, In y s' <-> ~ E.eq x y /\ In y s}.
Proof.
intros; exists (remove x s); intuition.
absurd (In x (remove x s)); auto.
@@ -74,66 +74,66 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
eauto.
Qed.
- Definition union
- forall s s' t, {s'' : t | forall x : elt, In x s'' <-> In x s \/ In x s'}.
+ Definition union :
+ forall s s' : t, {s'' : t | forall x : elt, In x s'' <-> In x s \/ In x s'}.
Proof.
intros; exists (union s s'); intuition.
Qed.
- Definition inter
- forall s s' t, {s'' : t | forall x : elt, In x s'' <-> In x s /\ In x s'}.
+ Definition inter :
+ forall s s' : t, {s'' : t | forall x : elt, In x s'' <-> In x s /\ In x s'}.
Proof.
intros; exists (inter s s'); intuition; eauto.
Qed.
- Definition diff
- forall s s' t, {s'' : t | forall x : elt, In x s'' <-> In x s /\ ~ In x s'}.
+ Definition diff :
+ forall s s' : t, {s'' : t | forall x : elt, In x s'' <-> In x s /\ ~ In x s'}.
Proof.
intros; exists (diff s s'); intuition; eauto.
absurd (In x s'); eauto.
Qed.
- Definition equal forall s s' : t, {Equal s s'} + {~ Equal s s'}.
+ Definition equal : forall s s' : t, {Equal s s'} + {~ Equal s s'}.
Proof.
intros.
- generalize (equal_1 (s=s) (s':=s')) (equal_2 (s:=s) (s':=s')).
+ generalize (equal_1 (s:=s) (s':=s')) (equal_2 (s:=s) (s':=s')).
case (equal s s'); intuition.
Qed.
- Definition subset forall s s' : t, {Subset s s'} + {~Subset s s'}.
+ Definition subset : forall s s' : t, {Subset s s'} + {~Subset s s'}.
Proof.
intros.
- generalize (subset_1 (s=s) (s':=s')) (subset_2 (s:=s) (s':=s')).
+ generalize (subset_1 (s:=s) (s':=s')) (subset_2 (s:=s) (s':=s')).
case (subset s s'); intuition.
Qed.
- Definition elements
- forall s t,
- {l list elt | ME.Sort l /\ (forall x : elt, In x s <-> ME.In x l)}.
+ Definition elements :
+ forall s : t,
+ {l : list elt | ME.Sort l /\ (forall x : elt, In x s <-> ME.In x l)}.
Proof.
intros; exists (elements s); intuition.
Defined.
- Definition fold
- forall (A Set) (f : elt -> A -> A) (s : t) (i : A),
- {r A | let (l,_) := elements s in
+ Definition fold :
+ forall (A : Set) (f : elt -> A -> A) (s : t) (i : A),
+ {r : A | let (l,_) := elements s in
r = fold_left (fun a e => f e a) l i}.
Proof.
- intros; exists (fold (A=A) f s i); exact (fold_1 s i f).
+ intros; exists (fold (A:=A) f s i); exact (fold_1 s i f).
Qed.
- Definition cardinal
- forall s t,
- {r nat | let (l,_) := elements s in r = length l }.
+ Definition cardinal :
+ forall s : t,
+ {r : nat | let (l,_) := elements s in r = length l }.
Proof.
intros; exists (cardinal s); exact (cardinal_1 s).
Qed.
- Definition fdec (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
- (x elt) := if Pdec x then true else false.
+ Definition fdec (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
+ (x : elt) := if Pdec x then true else false.
- Lemma compat_P_aux
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}),
+ Lemma compat_P_aux :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}),
compat_P E.eq P -> compat_bool E.eq (fdec Pdec).
Proof.
unfold compat_P, compat_bool, fdec in |- *; intros.
@@ -142,9 +142,9 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
Hint Resolve compat_P_aux.
- Definition filter
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
- {s' t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.
+ Definition filter :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+ {s' : t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.
Proof.
intros.
exists (filter (fdec Pdec) s).
@@ -160,13 +160,13 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
case (Pdec x); intuition.
Qed.
- Definition for_all
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+ Definition for_all :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
{compat_P E.eq P -> For_all P s} + {compat_P E.eq P -> ~ For_all P s}.
Proof.
intros.
- generalize (for_all_1 (s=s) (f:=fdec Pdec))
- (for_all_2 (s=s) (f:=fdec Pdec)).
+ generalize (for_all_1 (s:=s) (f:=fdec Pdec))
+ (for_all_2 (s:=s) (f:=fdec Pdec)).
case (for_all (fdec Pdec) s); unfold For_all in |- *; [ left | right ];
intros.
assert (compat_bool E.eq (fdec Pdec)); auto.
@@ -181,13 +181,13 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
case (Pdec x); intuition.
Qed.
- Definition exists_
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+ Definition exists_ :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
{compat_P E.eq P -> Exists P s} + {compat_P E.eq P -> ~ Exists P s}.
Proof.
intros.
- generalize (exists_1 (s=s) (f:=fdec Pdec))
- (exists_2 (s=s) (f:=fdec Pdec)).
+ generalize (exists_1 (s:=s) (f:=fdec Pdec))
+ (exists_2 (s:=s) (f:=fdec Pdec)).
case (exists_ (fdec Pdec) s); unfold Exists in |- *; [ left | right ];
intros.
elim H0; auto; intros.
@@ -204,18 +204,18 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
case (Pdec x); intuition.
Qed.
- Definition partition
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
- {partition t * t |
- let (s1, s2) = partition in
+ Definition partition :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+ {partition : t * t |
+ let (s1, s2) := partition in
compat_P E.eq P ->
For_all P s1 /\
For_all (fun x => ~ P x) s2 /\
- (forall x elt, In x s <-> In x s1 \/ In x s2)}.
+ (forall x : elt, In x s <-> In x s1 \/ In x s2)}.
Proof.
intros.
exists (partition (fdec Pdec) s).
- generalize (partition_1 s (f=fdec Pdec)) (partition_2 s (f:=fdec Pdec)).
+ generalize (partition_1 s (f:=fdec Pdec)) (partition_2 s (f:=fdec Pdec)).
case (partition (fdec Pdec) s).
intros s1 s2; simpl in |- *.
intros; assert (compat_bool E.eq (fdec Pdec)); auto.
@@ -234,196 +234,196 @@ Module DepOfNodep (M S) <: Sdep with Module E := M.E.
cut ((fun x => negb (fdec Pdec x)) x = true).
unfold fdec in |- *; case (Pdec x); intuition.
change ((fun x => negb (fdec Pdec x)) x = true) in |- *.
- apply (filter_2 (s=s) (x:=x)); auto.
- set (b = fdec Pdec x) in *; generalize (refl_equal b);
+ apply (filter_2 (s:=s) (x:=x)); auto.
+ set (b := fdec Pdec x) in *; generalize (refl_equal b);
pattern b at -1 in |- *; case b; unfold b in |- *;
[ left | right ].
elim (H4 x); intros _ B; apply B; auto.
elim (H x); intros _ B; apply B; auto.
apply filter_3; auto.
rewrite H5; auto.
- eapply (filter_1 (s=s) (x:=x) H2); elim (H4 x); intros B _; apply B;
+ eapply (filter_1 (s:=s) (x:=x) H2); elim (H4 x); intros B _; apply B;
auto.
- eapply (filter_1 (s=s) (x:=x) H3); elim (H x); intros B _; apply B; auto.
+ eapply (filter_1 (s:=s) (x:=x) H3); elim (H x); intros B _; apply B; auto.
Qed.
- Definition choose forall s : t, {x : elt | In x s} + {Empty s}.
+ Definition choose : forall s : t, {x : elt | In x s} + {Empty s}.
Proof.
intros.
- generalize (choose_1 (s=s)) (choose_2 (s:=s)).
+ generalize (choose_1 (s:=s)) (choose_2 (s:=s)).
case (choose s); [ left | right ]; auto.
exists e; auto.
Qed.
- Definition min_elt
- forall s t,
- {x elt | In x s /\ For_all (fun y => ~ E.lt y x) s} + {Empty s}.
+ Definition min_elt :
+ forall s : t,
+ {x : elt | In x s /\ For_all (fun y => ~ E.lt y x) s} + {Empty s}.
Proof.
intros;
- generalize (min_elt_1 (s=s)) (min_elt_2 (s:=s)) (min_elt_3 (s:=s)).
+ generalize (min_elt_1 (s:=s)) (min_elt_2 (s:=s)) (min_elt_3 (s:=s)).
case (min_elt s); [ left | right ]; auto.
exists e; unfold For_all in |- *; eauto.
Qed.
- Definition max_elt
- forall s t,
- {x elt | In x s /\ For_all (fun y => ~ E.lt x y) s} + {Empty s}.
+ Definition max_elt :
+ forall s : t,
+ {x : elt | In x s /\ For_all (fun y => ~ E.lt x y) s} + {Empty s}.
Proof.
intros;
- generalize (max_elt_1 (s=s)) (max_elt_2 (s:=s)) (max_elt_3 (s:=s)).
+ generalize (max_elt_1 (s:=s)) (max_elt_2 (s:=s)) (max_elt_3 (s:=s)).
case (max_elt s); [ left | right ]; auto.
exists e; unfold For_all in |- *; eauto.
Qed.
- Module E = E.
+ Module E := E.
- Definition elt = elt.
- Definition t = t.
+ Definition elt := elt.
+ Definition t := t.
- Definition In = In.
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) (s : t) :=
- forall x elt, In x s -> P x.
- Definition Exists (P elt -> Prop) (s : t) :=
- exists x elt, In x s /\ P x.
+ Definition In := In.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) (s : t) :=
+ forall x : elt, In x s -> P x.
+ Definition Exists (P : elt -> Prop) (s : t) :=
+ exists x : elt, In x s /\ P x.
- Definition eq_In = In_1.
+ Definition eq_In := In_1.
- Definition eq = Equal.
- Definition lt = lt.
- Definition eq_refl = eq_refl.
- Definition eq_sym = eq_sym.
- Definition eq_trans = eq_trans.
- Definition lt_trans = lt_trans.
- Definition lt_not_eq = lt_not_eq.
- Definition compare = compare.
+ Definition eq := Equal.
+ Definition lt := lt.
+ Definition eq_refl := eq_refl.
+ Definition eq_sym := eq_sym.
+ Definition eq_trans := eq_trans.
+ Definition lt_trans := lt_trans.
+ Definition lt_not_eq := lt_not_eq.
+ Definition compare := compare.
End DepOfNodep.
(** * From dependent signature [Sdep] to non-dependent signature [S]. *)
-Module NodepOfDep (M Sdep) <: S with Module E := M.E.
+Module NodepOfDep (M: Sdep) <: S with Module E := M.E.
Import M.
- Module ME = OrderedTypeFacts E.
+ Module ME := OrderedTypeFacts E.
- Definition empty t := let (s, _) := empty in s.
+ Definition empty : t := let (s, _) := empty in s.
- Lemma empty_1 Empty empty.
+ Lemma empty_1 : Empty empty.
Proof.
unfold empty in |- *; case M.empty; auto.
Qed.
- Definition is_empty (s t) : bool :=
+ Definition is_empty (s : t) : bool :=
if is_empty s then true else false.
- Lemma is_empty_1 forall s : t, Empty s -> is_empty s = true.
+ Lemma is_empty_1 : forall s : t, Empty s -> is_empty s = true.
Proof.
intros; unfold is_empty in |- *; case (M.is_empty s); auto.
Qed.
- Lemma is_empty_2 forall s : t, is_empty s = true -> Empty s.
+ Lemma is_empty_2 : forall s : t, is_empty s = true -> Empty s.
Proof.
intro s; unfold is_empty in |- *; case (M.is_empty s); auto.
intros; discriminate H.
Qed.
- Definition mem (x elt) (s : t) : bool :=
+ Definition mem (x : elt) (s : t) : bool :=
if mem x s then true else false.
- Lemma mem_1 forall (s : t) (x : elt), In x s -> mem x s = true.
+ Lemma mem_1 : forall (s : t) (x : elt), In x s -> mem x s = true.
Proof.
intros; unfold mem in |- *; case (M.mem x s); auto.
Qed.
- Lemma mem_2 forall (s : t) (x : elt), mem x s = true -> In x s.
+ Lemma mem_2 : forall (s : t) (x : elt), mem x s = true -> In x s.
Proof.
intros s x; unfold mem in |- *; case (M.mem x s); auto.
intros; discriminate H.
Qed.
- Definition equal (s s' t) : bool :=
+ Definition equal (s s' : t) : bool :=
if equal s s' then true else false.
- Lemma equal_1 forall s s' : t, Equal s s' -> equal s s' = true.
+ Lemma equal_1 : forall s s' : t, Equal s s' -> equal s s' = true.
Proof.
intros; unfold equal in |- *; case M.equal; intuition.
Qed.
- Lemma equal_2 forall s s' : t, equal s s' = true -> Equal s s'.
+ Lemma equal_2 : forall s s' : t, equal s s' = true -> Equal s s'.
Proof.
intros s s'; unfold equal in |- *; case (M.equal s s'); intuition;
inversion H.
Qed.
- Definition subset (s s' t) : bool :=
+ Definition subset (s s' : t) : bool :=
if subset s s' then true else false.
- Lemma subset_1 forall s s' : t, Subset s s' -> subset s s' = true.
+ Lemma subset_1 : forall s s' : t, Subset s s' -> subset s s' = true.
Proof.
intros; unfold subset in |- *; case M.subset; intuition.
Qed.
- Lemma subset_2 forall s s' : t, subset s s' = true -> Subset s s'.
+ Lemma subset_2 : forall s s' : t, subset s s' = true -> Subset s s'.
Proof.
intros s s'; unfold subset in |- *; case (M.subset s s'); intuition;
inversion H.
Qed.
- Definition choose (s t) : option elt :=
+ Definition choose (s : t) : option elt :=
match choose s with
| inleft (exist x _) => Some x
| inright _ => None
end.
- Lemma choose_1 forall (s : t) (x : elt), choose s = Some x -> In x s.
+ Lemma choose_1 : forall (s : t) (x : elt), choose s = Some x -> In x s.
Proof.
intros s x; unfold choose in |- *; case (M.choose s).
simple destruct s0; intros; injection H; intros; subst; auto.
intros; discriminate H.
Qed.
- Lemma choose_2 forall s : t, choose s = None -> Empty s.
+ Lemma choose_2 : forall s : t, choose s = None -> Empty s.
Proof.
intro s; unfold choose in |- *; case (M.choose s); auto.
simple destruct s0; intros; discriminate H.
Qed.
- Definition elements (s t) : list elt := let (l, _) := elements s in l.
+ Definition elements (s : t) : list elt := let (l, _) := elements s in l.
- Lemma elements_1 forall (s : t) (x : elt), In x s -> ME.In x (elements s).
+ Lemma elements_1 : forall (s : t) (x : elt), In x s -> ME.In x (elements s).
Proof.
intros; unfold elements in |- *; case (M.elements s); firstorder.
Qed.
- Lemma elements_2 forall (s : t) (x : elt), ME.In x (elements s) -> In x s.
+ Lemma elements_2 : forall (s : t) (x : elt), ME.In x (elements s) -> In x s.
Proof.
intros s x; unfold elements in |- *; case (M.elements s); firstorder.
Qed.
- Lemma elements_3 forall s : t, ME.Sort (elements s).
+ Lemma elements_3 : forall s : t, ME.Sort (elements s).
Proof.
intros; unfold elements in |- *; case (M.elements s); firstorder.
Qed.
- Definition min_elt (s t) : option elt :=
+ Definition min_elt (s : t) : option elt :=
match min_elt s with
| inleft (exist x _) => Some x
| inright _ => None
end.
- Lemma min_elt_1 forall (s : t) (x : elt), min_elt s = Some x -> In x s.
+ Lemma min_elt_1 : forall (s : t) (x : elt), min_elt s = Some x -> In x s.
Proof.
intros s x; unfold min_elt in |- *; case (M.min_elt s).
simple destruct s0; intros; injection H; intros; subst; intuition.
intros; discriminate H.
Qed.
- Lemma min_elt_2
- forall (s t) (x y : elt), min_elt s = Some x -> In y s -> ~ E.lt y x.
+ Lemma min_elt_2 :
+ forall (s : t) (x y : elt), min_elt s = Some x -> In y s -> ~ E.lt y x.
Proof.
intros s x y; unfold min_elt in |- *; case (M.min_elt s).
unfold For_all in |- *; simple destruct s0; intros; injection H; intros;
@@ -431,27 +431,27 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
intros; discriminate H.
Qed.
- Lemma min_elt_3 forall s : t, min_elt s = None -> Empty s.
+ Lemma min_elt_3 : forall s : t, min_elt s = None -> Empty s.
Proof.
intros s; unfold min_elt in |- *; case (M.min_elt s); auto.
simple destruct s0; intros; discriminate H.
Qed.
- Definition max_elt (s t) : option elt :=
+ Definition max_elt (s : t) : option elt :=
match max_elt s with
| inleft (exist x _) => Some x
| inright _ => None
end.
- Lemma max_elt_1 forall (s : t) (x : elt), max_elt s = Some x -> In x s.
+ Lemma max_elt_1 : forall (s : t) (x : elt), max_elt s = Some x -> In x s.
Proof.
intros s x; unfold max_elt in |- *; case (M.max_elt s).
simple destruct s0; intros; injection H; intros; subst; intuition.
intros; discriminate H.
Qed.
- Lemma max_elt_2
- forall (s t) (x y : elt), max_elt s = Some x -> In y s -> ~ E.lt x y.
+ Lemma max_elt_2 :
+ forall (s : t) (x y : elt), max_elt s = Some x -> In y s -> ~ E.lt x y.
Proof.
intros s x y; unfold max_elt in |- *; case (M.max_elt s).
unfold For_all in |- *; simple destruct s0; intros; injection H; intros;
@@ -459,144 +459,144 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
intros; discriminate H.
Qed.
- Lemma max_elt_3 forall s : t, max_elt s = None -> Empty s.
+ Lemma max_elt_3 : forall s : t, max_elt s = None -> Empty s.
Proof.
intros s; unfold max_elt in |- *; case (M.max_elt s); auto.
simple destruct s0; intros; discriminate H.
Qed.
- Definition add (x elt) (s : t) : t := let (s', _) := add x s in s'.
+ Definition add (x : elt) (s : t) : t := let (s', _) := add x s in s'.
- Lemma add_1 forall (s : t) (x y : elt), E.eq x y -> In y (add x s).
+ Lemma add_1 : forall (s : t) (x y : elt), E.eq x y -> In y (add x s).
Proof.
intros; unfold add in |- *; case (M.add x s); unfold Add in |- *;
firstorder.
Qed.
- Lemma add_2 forall (s : t) (x y : elt), In y s -> In y (add x s).
+ Lemma add_2 : forall (s : t) (x y : elt), In y s -> In y (add x s).
Proof.
intros; unfold add in |- *; case (M.add x s); unfold Add in |- *;
firstorder.
Qed.
- Lemma add_3
- forall (s t) (x y : elt), ~ E.eq x y -> In y (add x s) -> In y s.
+ Lemma add_3 :
+ forall (s : t) (x y : elt), ~ E.eq x y -> In y (add x s) -> In y s.
Proof.
intros s x y; unfold add in |- *; case (M.add x s); unfold Add in |- *;
firstorder.
Qed.
- Definition remove (x elt) (s : t) : t := let (s', _) := remove x s in s'.
+ Definition remove (x : elt) (s : t) : t := let (s', _) := remove x s in s'.
- Lemma remove_1 forall (s : t) (x y : elt), E.eq x y -> ~ In y (remove x s).
+ Lemma remove_1 : forall (s : t) (x y : elt), E.eq x y -> ~ In y (remove x s).
Proof.
intros; unfold remove in |- *; case (M.remove x s); firstorder.
Qed.
- Lemma remove_2
- forall (s t) (x y : elt), ~ E.eq x y -> In y s -> In y (remove x s).
+ Lemma remove_2 :
+ forall (s : t) (x y : elt), ~ E.eq x y -> In y s -> In y (remove x s).
Proof.
intros; unfold remove in |- *; case (M.remove x s); firstorder.
Qed.
- Lemma remove_3 forall (s : t) (x y : elt), In y (remove x s) -> In y s.
+ Lemma remove_3 : forall (s : t) (x y : elt), In y (remove x s) -> In y s.
Proof.
intros s x y; unfold remove in |- *; case (M.remove x s); firstorder.
Qed.
- Definition singleton (x elt) : t := let (s, _) := singleton x in s.
+ Definition singleton (x : elt) : t := let (s, _) := singleton x in s.
- Lemma singleton_1 forall x y : elt, In y (singleton x) -> E.eq x y.
+ Lemma singleton_1 : forall x y : elt, In y (singleton x) -> E.eq x y.
Proof.
intros x y; unfold singleton in |- *; case (M.singleton x); firstorder.
Qed.
- Lemma singleton_2 forall x y : elt, E.eq x y -> In y (singleton x).
+ Lemma singleton_2 : forall x y : elt, E.eq x y -> In y (singleton x).
Proof.
intros x y; unfold singleton in |- *; case (M.singleton x); firstorder.
Qed.
- Definition union (s s' t) : t := let (s'', _) := union s s' in s''.
+ Definition union (s s' : t) : t := let (s'', _) := union s s' in s''.
- Lemma union_1
- forall (s s' t) (x : elt), In x (union s s') -> In x s \/ In x s'.
+ Lemma union_1 :
+ forall (s s' : t) (x : elt), In x (union s s') -> In x s \/ In x s'.
Proof.
intros s s' x; unfold union in |- *; case (M.union s s'); firstorder.
Qed.
- Lemma union_2 forall (s s' : t) (x : elt), In x s -> In x (union s s').
+ Lemma union_2 : forall (s s' : t) (x : elt), In x s -> In x (union s s').
Proof.
intros s s' x; unfold union in |- *; case (M.union s s'); firstorder.
Qed.
- Lemma union_3 forall (s s' : t) (x : elt), In x s' -> In x (union s s').
+ Lemma union_3 : forall (s s' : t) (x : elt), In x s' -> In x (union s s').
Proof.
intros s s' x; unfold union in |- *; case (M.union s s'); firstorder.
Qed.
- Definition inter (s s' t) : t := let (s'', _) := inter s s' in s''.
+ Definition inter (s s' : t) : t := let (s'', _) := inter s s' in s''.
- Lemma inter_1 forall (s s' : t) (x : elt), In x (inter s s') -> In x s.
+ Lemma inter_1 : forall (s s' : t) (x : elt), In x (inter s s') -> In x s.
Proof.
intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder.
Qed.
- Lemma inter_2 forall (s s' : t) (x : elt), In x (inter s s') -> In x s'.
+ Lemma inter_2 : forall (s s' : t) (x : elt), In x (inter s s') -> In x s'.
Proof.
intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder.
Qed.
- Lemma inter_3
- forall (s s' t) (x : elt), In x s -> In x s' -> In x (inter s s').
+ Lemma inter_3 :
+ forall (s s' : t) (x : elt), In x s -> In x s' -> In x (inter s s').
Proof.
intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder.
Qed.
- Definition diff (s s' t) : t := let (s'', _) := diff s s' in s''.
+ Definition diff (s s' : t) : t := let (s'', _) := diff s s' in s''.
- Lemma diff_1 forall (s s' : t) (x : elt), In x (diff s s') -> In x s.
+ Lemma diff_1 : forall (s s' : t) (x : elt), In x (diff s s') -> In x s.
Proof.
intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder.
Qed.
- Lemma diff_2 forall (s s' : t) (x : elt), In x (diff s s') -> ~ In x s'.
+ Lemma diff_2 : forall (s s' : t) (x : elt), In x (diff s s') -> ~ In x s'.
Proof.
intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder.
Qed.
- Lemma diff_3
- forall (s s' t) (x : elt), In x s -> ~ In x s' -> In x (diff s s').
+ Lemma diff_3 :
+ forall (s s' : t) (x : elt), In x s -> ~ In x s' -> In x (diff s s').
Proof.
intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder.
Qed.
- Definition cardinal (s t) : nat := let (f, _) := cardinal s in f.
+ Definition cardinal (s : t) : nat := let (f, _) := cardinal s in f.
- Lemma cardinal_1 forall s, cardinal s = length (elements s).
+ Lemma cardinal_1 : forall s, cardinal s = length (elements s).
Proof.
intros; unfold cardinal in |- *; case (M.cardinal s); unfold elements in *;
destruct (M.elements s); auto.
Qed.
- Definition fold (B Set) (f : elt -> B -> B) (i : t)
- (s B) : B := let (fold, _) := fold f i s in fold.
+ Definition fold (B : Set) (f : elt -> B -> B) (i : t)
+ (s : B) : B := let (fold, _) := fold f i s in fold.
- Lemma fold_1
- forall (s t) (A : Set) (i : A) (f : elt -> A -> A),
+ Lemma fold_1 :
+ forall (s : t) (A : Set) (i : A) (f : elt -> A -> A),
fold f s i = fold_left (fun a e => f e a) (elements s) i.
Proof.
intros; unfold fold in |- *; case (M.fold f s i); unfold elements in *;
destruct (M.elements s); auto.
Qed.
- Definition f_dec
- forall (f elt -> bool) (x : elt), {f x = true} + {f x <> true}.
+ Definition f_dec :
+ forall (f : elt -> bool) (x : elt), {f x = true} + {f x <> true}.
Proof.
intros; case (f x); auto with bool.
Defined.
- Lemma compat_P_aux
- forall f elt -> bool,
+ Lemma compat_P_aux :
+ forall f : elt -> bool,
compat_bool E.eq f -> compat_P E.eq (fun x => f x = true).
Proof.
unfold compat_bool, compat_P in |- *; intros; rewrite <- H1; firstorder.
@@ -604,40 +604,40 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
Hint Resolve compat_P_aux.
- Definition filter (f elt -> bool) (s : t) : t :=
- let (s', _) = filter (P:=fun x => f x = true) (f_dec f) s in s'.
+ Definition filter (f : elt -> bool) (s : t) : t :=
+ let (s', _) := filter (P:=fun x => f x = true) (f_dec f) s in s'.
- Lemma filter_1
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_1 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool E.eq f -> In x (filter f s) -> In x s.
Proof.
intros s x f; unfold filter in |- *; case M.filter; intuition.
generalize (i (compat_P_aux H)); firstorder.
Qed.
- Lemma filter_2
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_2 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool E.eq f -> In x (filter f s) -> f x = true.
Proof.
intros s x f; unfold filter in |- *; case M.filter; intuition.
generalize (i (compat_P_aux H)); firstorder.
Qed.
- Lemma filter_3
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_3 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s).
Proof.
intros s x f; unfold filter in |- *; case M.filter; intuition.
generalize (i (compat_P_aux H)); firstorder.
Qed.
- Definition for_all (f elt -> bool) (s : t) : bool :=
- if for_all (P=fun x => f x = true) (f_dec f) s
+ Definition for_all (f : elt -> bool) (s : t) : bool :=
+ if for_all (P:=fun x => f x = true) (f_dec f) s
then true
else false.
- Lemma for_all_1
- forall (s t) (f : elt -> bool),
+ Lemma for_all_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool E.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
Proof.
@@ -645,8 +645,8 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
auto.
Qed.
- Lemma for_all_2
- forall (s t) (f : elt -> bool),
+ Lemma for_all_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool E.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.
Proof.
@@ -654,33 +654,33 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
inversion H0.
Qed.
- Definition exists_ (f elt -> bool) (s : t) : bool :=
- if exists_ (P=fun x => f x = true) (f_dec f) s
+ Definition exists_ (f : elt -> bool) (s : t) : bool :=
+ if exists_ (P:=fun x => f x = true) (f_dec f) s
then true
else false.
- Lemma exists_1
- forall (s t) (f : elt -> bool),
+ Lemma exists_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool E.eq f -> Exists (fun x => f x = true) s -> exists_ f s = true.
Proof.
intros s f; unfold exists_ in |- *; case M.exists_; intuition; elim n;
auto.
Qed.
- Lemma exists_2
- forall (s t) (f : elt -> bool),
+ Lemma exists_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool E.eq f -> exists_ f s = true -> Exists (fun x => f x = true) s.
Proof.
intros s f; unfold exists_ in |- *; case M.exists_; intuition;
inversion H0.
Qed.
- Definition partition (f elt -> bool) (s : t) :
- t * t =
- let (p, _) = partition (P:=fun x => f x = true) (f_dec f) s in p.
+ Definition partition (f : elt -> bool) (s : t) :
+ t * t :=
+ let (p, _) := partition (P:=fun x => f x = true) (f_dec f) s in p.
- Lemma partition_1
- forall (s t) (f : elt -> bool),
+ Lemma partition_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool E.eq f -> Equal (fst (partition f s)) (filter f s).
Proof.
intros s f; unfold partition in |- *; case M.partition.
@@ -697,14 +697,14 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
eapply filter_2; eauto.
Qed.
- Lemma partition_2
- forall (s t) (f : elt -> bool),
+ Lemma partition_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool E.eq f -> Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
Proof.
intros s f; unfold partition in |- *; case M.partition.
intro p; case p; clear p; intros s1 s2 H C.
generalize (H (compat_P_aux C)); clear H; intro H.
- assert (D compat_bool E.eq (fun x => negb (f x))).
+ assert (D : compat_bool E.eq (fun x => negb (f x))).
generalize C; unfold compat_bool in |- *; intros; apply (f_equal negb);
auto.
simpl in |- *; unfold Equal in |- *; intuition.
@@ -721,30 +721,30 @@ Module NodepOfDep (M Sdep) <: S with Module E := M.E.
Qed.
- Module E = E.
- Definition elt = elt.
- Definition t = t.
-
- Definition In = In.
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Add (x elt) (s s' : t) :=
- forall y elt, In y s' <-> E.eq y x \/ In y s.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) (s : t) :=
- forall x elt, In x s -> P x.
- Definition Exists (P elt -> Prop) (s : t) :=
- exists x elt, In x s /\ P x.
-
- Definition In_1 = eq_In.
-
- Definition eq = Equal.
- Definition lt = lt.
- Definition eq_refl = eq_refl.
- Definition eq_sym = eq_sym.
- Definition eq_trans = eq_trans.
- Definition lt_trans = lt_trans.
- Definition lt_not_eq = lt_not_eq.
- Definition compare = compare.
+ Module E := E.
+ Definition elt := elt.
+ Definition t := t.
+
+ Definition In := In.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Add (x : elt) (s s' : t) :=
+ forall y : elt, In y s' <-> E.eq y x \/ In y s.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) (s : t) :=
+ forall x : elt, In x s -> P x.
+ Definition Exists (P : elt -> Prop) (s : t) :=
+ exists x : elt, In x s /\ P x.
+
+ Definition In_1 := eq_In.
+
+ Definition eq := Equal.
+ Definition lt := lt.
+ Definition eq_refl := eq_refl.
+ Definition eq_sym := eq_sym.
+ Definition eq_trans := eq_trans.
+ Definition lt_trans := lt_trans.
+ Definition lt_not_eq := lt_not_eq.
+ Definition compare := compare.
End NodepOfDep.
diff --git a/theories/FSets/FSetEqProperties.v b/theories/FSets/FSetEqProperties.v
index 2adc17ff4..2abc20799 100644
--- a/theories/FSets/FSetEqProperties.v
+++ b/theories/FSets/FSetEqProperties.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetEqProperties.v,v 1.10 2006/03/14 23:52:28 letouzey Exp $ *)
(** * Finite sets library *)
@@ -23,118 +23,118 @@ Require Import Zerob.
Require Import Sumbool.
Require Import Omega.
-Module EqProperties (MS).
+Module EqProperties (M:S).
Import M.
Import Logic. (* to unmask [eq] *)
Import Peano. (* to unmask [lt] *)
-Module ME = OrderedTypeFacts E.
-Module MP = Properties M.
+Module ME := OrderedTypeFacts E.
+Module MP := Properties M.
Import MP.
Import MP.FM.
-Definition Add = MP.Add.
+Definition Add := MP.Add.
Section BasicProperties.
(** Some old specifications written with boolean equalities. *)
-Variable s s' s'' t.
-Variable x y z elt.
+Variable s s' s'': t.
+Variable x y z : elt.
-Lemma mem_eq
+Lemma mem_eq:
E.eq x y -> mem x s=mem y s.
Proof.
intro H; rewrite H; auto.
Qed.
-Lemma equal_mem_1
+Lemma equal_mem_1:
(forall a, mem a s=mem a s') -> equal s s'=true.
Proof.
intros; apply equal_1; unfold Equal; intros.
do 2 rewrite mem_iff; rewrite H; tauto.
Qed.
-Lemma equal_mem_2
+Lemma equal_mem_2:
equal s s'=true -> forall a, mem a s=mem a s'.
Proof.
intros; rewrite (equal_2 H); auto.
Qed.
-Lemma subset_mem_1
+Lemma subset_mem_1:
(forall a, mem a s=true->mem a s'=true) -> subset s s'=true.
Proof.
intros; apply subset_1; unfold Subset; intros a.
do 2 rewrite mem_iff; auto.
Qed.
-Lemma subset_mem_2
+Lemma subset_mem_2:
subset s s'=true -> forall a, mem a s=true -> mem a s'=true.
Proof.
intros H a; do 2 rewrite <- mem_iff; apply subset_2; auto.
Qed.
-Lemma empty_mem mem x empty=false.
+Lemma empty_mem: mem x empty=false.
Proof.
rewrite <- not_mem_iff; auto.
Qed.
-Lemma is_empty_equal_empty is_empty s = equal s empty.
+Lemma is_empty_equal_empty: is_empty s = equal s empty.
Proof.
apply bool_1; split; intros.
-rewrite <- (empty_is_empty_1 (s=empty)); auto with set.
+rewrite <- (empty_is_empty_1 (s:=empty)); auto with set.
rewrite <- is_empty_iff; auto with set.
Qed.
-Lemma choose_mem_1 choose s=Some x -> mem x s=true.
+Lemma choose_mem_1: choose s=Some x -> mem x s=true.
Proof.
auto.
Qed.
-Lemma choose_mem_2 choose s=None -> is_empty s=true.
+Lemma choose_mem_2: choose s=None -> is_empty s=true.
Proof.
auto.
Qed.
-Lemma add_mem_1 mem x (add x s)=true.
+Lemma add_mem_1: mem x (add x s)=true.
Proof.
auto.
Qed.
-Lemma add_mem_2 ~E.eq x y -> mem y (add x s)=mem y s.
+Lemma add_mem_2: ~E.eq x y -> mem y (add x s)=mem y s.
Proof.
apply add_neq_b.
Qed.
-Lemma remove_mem_1 mem x (remove x s)=false.
+Lemma remove_mem_1: mem x (remove x s)=false.
Proof.
rewrite <- not_mem_iff; auto.
Qed.
-Lemma remove_mem_2 ~E.eq x y -> mem y (remove x s)=mem y s.
+Lemma remove_mem_2: ~E.eq x y -> mem y (remove x s)=mem y s.
Proof.
apply remove_neq_b.
Qed.
-Lemma singleton_equal_add
+Lemma singleton_equal_add:
equal (singleton x) (add x empty)=true.
Proof.
rewrite (singleton_equal_add x); auto with set.
Qed.
-Lemma union_mem
+Lemma union_mem:
mem x (union s s')=mem x s || mem x s'.
Proof.
apply union_b.
Qed.
-Lemma inter_mem
+Lemma inter_mem:
mem x (inter s s')=mem x s && mem x s'.
Proof.
apply inter_b.
Qed.
-Lemma diff_mem
+Lemma diff_mem:
mem x (diff s s')=mem x s && negb (mem x s').
Proof.
apply diff_b.
@@ -142,41 +142,41 @@ Qed.
(** properties of [mem] *)
-Lemma mem_3 ~In x s -> mem x s=false.
+Lemma mem_3 : ~In x s -> mem x s=false.
Proof.
intros; rewrite <- not_mem_iff; auto.
Qed.
-Lemma mem_4 mem x s=false -> ~In x s.
+Lemma mem_4 : mem x s=false -> ~In x s.
Proof.
intros; rewrite not_mem_iff; auto.
Qed.
(** Properties of [equal] *)
-Lemma equal_refl equal s s=true.
+Lemma equal_refl: equal s s=true.
Proof.
auto with set.
Qed.
-Lemma equal_sym equal s s'=equal s' s.
+Lemma equal_sym: equal s s'=equal s' s.
Proof.
intros; apply bool_1; do 2 rewrite <- equal_iff; intuition.
Qed.
-Lemma equal_trans
+Lemma equal_trans:
equal s s'=true -> equal s' s''=true -> equal s s''=true.
Proof.
intros; rewrite (equal_2 H); auto.
Qed.
-Lemma equal_equal
+Lemma equal_equal:
equal s s'=true -> equal s s''=equal s' s''.
Proof.
intros; rewrite (equal_2 H); auto.
Qed.
-Lemma equal_cardinal
+Lemma equal_cardinal:
equal s s'=true -> cardinal s=cardinal s'.
Proof.
auto with set.
@@ -184,25 +184,25 @@ Qed.
(* Properties of [subset] *)
-Lemma subset_refl subset s s=true.
+Lemma subset_refl: subset s s=true.
Proof.
auto with set.
Qed.
-Lemma subset_antisym
+Lemma subset_antisym:
subset s s'=true -> subset s' s=true -> equal s s'=true.
Proof.
auto with set.
Qed.
-Lemma subset_trans
+Lemma subset_trans:
subset s s'=true -> subset s' s''=true -> subset s s''=true.
Proof.
do 3 rewrite <- subset_iff; intros.
apply subset_trans with s'; auto.
Qed.
-Lemma subset_equal
+Lemma subset_equal:
equal s s'=true -> subset s s'=true.
Proof.
auto with set.
@@ -210,8 +210,8 @@ Qed.
(** Properties of [choose] *)
-Lemma choose_mem_3
- is_empty s=false -> {xelt|choose s=Some x /\ mem x s=true}.
+Lemma choose_mem_3:
+ is_empty s=false -> {x:elt|choose s=Some x /\ mem x s=true}.
Proof.
intros.
generalize (@choose_1 s) (@choose_2 s).
@@ -221,7 +221,7 @@ generalize (H1 (refl_equal None)); clear H1.
intros; rewrite (is_empty_1 H1) in H; discriminate.
Qed.
-Lemma choose_mem_4 choose empty=None.
+Lemma choose_mem_4: choose empty=None.
Proof.
generalize (@choose_1 empty).
case (@choose empty);intros;auto.
@@ -230,13 +230,13 @@ Qed.
(** Properties of [add] *)
-Lemma add_mem_3
+Lemma add_mem_3:
mem y s=true -> mem y (add x s)=true.
Proof.
auto.
Qed.
-Lemma add_equal
+Lemma add_equal:
mem x s=true -> equal (add x s) s=true.
Proof.
auto with set.
@@ -244,26 +244,26 @@ Qed.
(** Properties of [remove] *)
-Lemma remove_mem_3
+Lemma remove_mem_3:
mem y (remove x s)=true -> mem y s=true.
Proof.
rewrite remove_b; intros H;destruct (andb_prop _ _ H); auto.
Qed.
-Lemma remove_equal
+Lemma remove_equal:
mem x s=false -> equal (remove x s) s=true.
Proof.
intros; apply equal_1; apply remove_equal.
rewrite not_mem_iff; auto.
Qed.
-Lemma add_remove
+Lemma add_remove:
mem x s=true -> equal (add x (remove x s)) s=true.
Proof.
intros; apply equal_1; apply add_remove; auto.
Qed.
-Lemma remove_add
+Lemma remove_add:
mem x s=false -> equal (remove x (add x s)) s=true.
Proof.
intros; apply equal_1; apply remove_add; auto.
@@ -272,7 +272,7 @@ Qed.
(** Properties of [is_empty] *)
-Lemma is_empty_cardinal is_empty s = zerob (cardinal s).
+Lemma is_empty_cardinal: is_empty s = zerob (cardinal s).
Proof.
intros; apply bool_1; split; intros.
rewrite cardinal_1; simpl; auto.
@@ -282,61 +282,61 @@ Qed.
(** Properties of [singleton] *)
-Lemma singleton_mem_1 mem x (singleton x)=true.
+Lemma singleton_mem_1: mem x (singleton x)=true.
Proof.
auto with set.
Qed.
-Lemma singleton_mem_2 ~E.eq x y -> mem y (singleton x)=false.
+Lemma singleton_mem_2: ~E.eq x y -> mem y (singleton x)=false.
Proof.
intros; rewrite singleton_b.
unfold ME.eqb; destruct (ME.eq_dec x y); intuition.
Qed.
-Lemma singleton_mem_3 mem y (singleton x)=true -> E.eq x y.
+Lemma singleton_mem_3: mem y (singleton x)=true -> E.eq x y.
Proof.
auto.
Qed.
(** Properties of [union] *)
-Lemma union_sym
+Lemma union_sym:
equal (union s s') (union s' s)=true.
Proof.
auto with set.
Qed.
-Lemma union_subset_equal
+Lemma union_subset_equal:
subset s s'=true -> equal (union s s') s'=true.
Proof.
auto with set.
Qed.
-Lemma union_equal_1
+Lemma union_equal_1:
equal s s'=true-> equal (union s s'') (union s' s'')=true.
Proof.
auto with set.
Qed.
-Lemma union_equal_2
+Lemma union_equal_2:
equal s' s''=true-> equal (union s s') (union s s'')=true.
Proof.
auto with set.
Qed.
-Lemma union_assoc
+Lemma union_assoc:
equal (union (union s s') s'') (union s (union s' s''))=true.
Proof.
auto with set.
Qed.
-Lemma add_union_singleton
+Lemma add_union_singleton:
equal (add x s) (union (singleton x) s)=true.
Proof.
auto with set.
Qed.
-Lemma union_add
+Lemma union_add:
equal (union (add x s) s') (add x (union s s'))=true.
Proof.
auto with set.
@@ -344,17 +344,17 @@ Qed.
(* caracterisation of [union] via [subset] *)
-Lemma union_subset_1 subset s (union s s')=true.
+Lemma union_subset_1: subset s (union s s')=true.
Proof.
auto with set.
Qed.
-Lemma union_subset_2 subset s' (union s s')=true.
+Lemma union_subset_2: subset s' (union s s')=true.
Proof.
auto with set.
Qed.
-Lemma union_subset_3
+Lemma union_subset_3:
subset s s''=true -> subset s' s''=true ->
subset (union s s') s''=true.
Proof.
@@ -363,54 +363,54 @@ Qed.
(** Properties of [inter] *)
-Lemma inter_sym equal (inter s s') (inter s' s)=true.
+Lemma inter_sym: equal (inter s s') (inter s' s)=true.
Proof.
auto with set.
Qed.
-Lemma inter_subset_equal
+Lemma inter_subset_equal:
subset s s'=true -> equal (inter s s') s=true.
Proof.
auto with set.
Qed.
-Lemma inter_equal_1
+Lemma inter_equal_1:
equal s s'=true -> equal (inter s s'') (inter s' s'')=true.
Proof.
auto with set.
Qed.
-Lemma inter_equal_2
+Lemma inter_equal_2:
equal s' s''=true -> equal (inter s s') (inter s s'')=true.
Proof.
auto with set.
Qed.
-Lemma inter_assoc
+Lemma inter_assoc:
equal (inter (inter s s') s'') (inter s (inter s' s''))=true.
Proof.
auto with set.
Qed.
-Lemma union_inter_1
+Lemma union_inter_1:
equal (inter (union s s') s'') (union (inter s s'') (inter s' s''))=true.
Proof.
auto with set.
Qed.
-Lemma union_inter_2
+Lemma union_inter_2:
equal (union (inter s s') s'') (inter (union s s'') (union s' s''))=true.
Proof.
auto with set.
Qed.
-Lemma inter_add_1 mem x s'=true ->
+Lemma inter_add_1: mem x s'=true ->
equal (inter (add x s) s') (add x (inter s s'))=true.
Proof.
auto with set.
Qed.
-Lemma inter_add_2 mem x s'=false ->
+Lemma inter_add_2: mem x s'=false ->
equal (inter (add x s) s') (inter s s')=true.
Proof.
intros; apply equal_1; apply inter_add_2.
@@ -419,17 +419,17 @@ Qed.
(* caracterisation of [union] via [subset] *)
-Lemma inter_subset_1 subset (inter s s') s=true.
+Lemma inter_subset_1: subset (inter s s') s=true.
Proof.
auto with set.
Qed.
-Lemma inter_subset_2 subset (inter s s') s'=true.
+Lemma inter_subset_2: subset (inter s s') s'=true.
Proof.
auto with set.
Qed.
-Lemma inter_subset_3
+Lemma inter_subset_3:
subset s'' s=true -> subset s'' s'=true ->
subset s'' (inter s s')=true.
Proof.
@@ -438,30 +438,30 @@ Qed.
(** Properties of [diff] *)
-Lemma diff_subset subset (diff s s') s=true.
+Lemma diff_subset: subset (diff s s') s=true.
Proof.
auto with set.
Qed.
-Lemma diff_subset_equal
+Lemma diff_subset_equal:
subset s s'=true -> equal (diff s s') empty=true.
Proof.
auto with set.
Qed.
-Lemma remove_inter_singleton
+Lemma remove_inter_singleton:
equal (remove x s) (diff s (singleton x))=true.
Proof.
auto with set.
Qed.
-Lemma diff_inter_empty
+Lemma diff_inter_empty:
equal (inter (diff s s') (inter s s')) empty=true.
Proof.
auto with set.
Qed.
-Lemma diff_inter_all
+Lemma diff_inter_all:
equal (union (diff s s') (inter s s')) s=true.
Proof.
auto with set.
@@ -471,16 +471,16 @@ End BasicProperties.
Hint Immediate empty_mem is_empty_equal_empty add_mem_1
remove_mem_1 singleton_equal_add union_mem inter_mem
- diff_mem equal_sym add_remove remove_add set.
+ diff_mem equal_sym add_remove remove_add : set.
Hint Resolve equal_mem_1 subset_mem_1 choose_mem_1
choose_mem_2 add_mem_2 remove_mem_2 equal_refl equal_equal
subset_refl subset_equal subset_antisym
- add_mem_3 add_equal remove_mem_3 remove_equal set.
+ add_mem_3 add_equal remove_mem_3 remove_equal : set.
(** General recursion principes based on [cardinal] *)
-Lemma cardinal_set_rec forall (P:t->Type),
+Lemma cardinal_set_rec: forall (P:t->Type),
(forall s s', equal s s'=true -> P s -> P s') ->
(forall s x, mem x s=false -> P s -> P (add x s)) ->
P empty -> forall n s, cardinal s=n -> P s.
@@ -493,7 +493,7 @@ apply equal_1; intro a; rewrite add_iff; rewrite (H1 a); tauto.
apply X0; auto with set; apply mem_3; auto.
Qed.
-Lemma set_rec forall (P:t->Type),
+Lemma set_rec: forall (P:t->Type),
(forall s s', equal s s'=true -> P s -> P s') ->
(forall s x, mem x s=false -> P s -> P (add x s)) ->
P empty -> forall s, P s.
@@ -503,7 +503,7 @@ Qed.
(** Properties of [fold] *)
-Lemma exclusive_set forall s s' x,
+Lemma exclusive_set : forall s s' x,
~In x s\/~In x s' <-> mem x s && mem x s'=false.
Proof.
intros; do 2 rewrite not_mem_iff.
@@ -511,53 +511,53 @@ destruct (mem x s); destruct (mem x s'); intuition.
Qed.
Section Fold.
-Variables (ASet)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).
-Variables (felt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).
-Variables (iA).
-Variables (s s't)(x:elt).
+Variables (A:Set)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).
+Variables (f:elt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).
+Variables (i:A).
+Variables (s s':t)(x:elt).
-Lemma fold_empty eqA (fold f empty i) i.
+Lemma fold_empty: eqA (fold f empty i) i.
Proof.
apply fold_empty; auto.
Qed.
-Lemma fold_equal
+Lemma fold_equal:
equal s s'=true -> eqA (fold f s i) (fold f s' i).
Proof.
-intros; apply fold_equal with (eqA=eqA); auto.
+intros; apply fold_equal with (eqA:=eqA); auto.
Qed.
-Lemma fold_add
+Lemma fold_add:
mem x s=false -> eqA (fold f (add x s) i) (f x (fold f s i)).
Proof.
-intros; apply fold_add with (eqA=eqA); auto.
+intros; apply fold_add with (eqA:=eqA); auto.
rewrite not_mem_iff; auto.
Qed.
-Lemma add_fold
+Lemma add_fold:
mem x s=true -> eqA (fold f (add x s) i) (fold f s i).
Proof.
-intros; apply add_fold with (eqA=eqA); auto.
+intros; apply add_fold with (eqA:=eqA); auto.
Qed.
-Lemma remove_fold_1
+Lemma remove_fold_1:
mem x s=true -> eqA (f x (fold f (remove x s) i)) (fold f s i).
Proof.
-intros; apply remove_fold_1 with (eqA=eqA); auto.
+intros; apply remove_fold_1 with (eqA:=eqA); auto.
Qed.
-Lemma remove_fold_2
+Lemma remove_fold_2:
mem x s=false -> eqA (fold f (remove x s) i) (fold f s i).
Proof.
-intros; apply remove_fold_2 with (eqA=eqA); auto.
+intros; apply remove_fold_2 with (eqA:=eqA); auto.
rewrite not_mem_iff; auto.
Qed.
-Lemma fold_union
+Lemma fold_union:
(forall x, mem x s && mem x s'=false) ->
eqA (fold f (union s s') i) (fold f s (fold f s' i)).
Proof.
-intros; apply fold_union with (eqA=eqA); auto.
+intros; apply fold_union with (eqA:=eqA); auto.
intros; rewrite exclusive_set; auto.
Qed.
@@ -565,32 +565,32 @@ End Fold.
(** Properties of [cardinal] *)
-Lemma add_cardinal_1
+Lemma add_cardinal_1:
forall s x, mem x s=true -> cardinal (add x s)=cardinal s.
Proof.
auto with set.
Qed.
-Lemma add_cardinal_2
+Lemma add_cardinal_2:
forall s x, mem x s=false -> cardinal (add x s)=S (cardinal s).
Proof.
intros; apply add_cardinal_2; auto.
rewrite not_mem_iff; auto.
Qed.
-Lemma remove_cardinal_1
+Lemma remove_cardinal_1:
forall s x, mem x s=true -> S (cardinal (remove x s))=cardinal s.
Proof.
intros; apply remove_cardinal_1; auto.
Qed.
-Lemma remove_cardinal_2
+Lemma remove_cardinal_2:
forall s x, mem x s=false -> cardinal (remove x s)=cardinal s.
Proof.
auto with set.
Qed.
-Lemma union_cardinal
+Lemma union_cardinal:
forall s s', (forall x, mem x s && mem x s'=false) ->
cardinal (union s s')=cardinal s+cardinal s'.
Proof.
@@ -598,7 +598,7 @@ intros; apply union_cardinal; auto; intros.
rewrite exclusive_set; auto.
Qed.
-Lemma subset_cardinal
+Lemma subset_cardinal:
forall s s', subset s s'=true -> cardinal s<=cardinal s'.
Proof.
intros; apply subset_cardinal; auto.
@@ -608,20 +608,20 @@ Section Bool.
(** Properties of [filter] *)
-Variable felt->bool.
-Variable Comp compat_bool E.eq f.
+Variable f:elt->bool.
+Variable Comp: compat_bool E.eq f.
-Let Comp' compat_bool E.eq (fun x =>negb (f x)).
+Let Comp' : compat_bool E.eq (fun x =>negb (f x)).
Proof.
unfold compat_bool in *; intros; f_equal; auto.
Qed.
-Lemma filter_mem forall s x, mem x (filter f s)=mem x s && f x.
+Lemma filter_mem: forall s x, mem x (filter f s)=mem x s && f x.
Proof.
intros; apply filter_b; auto.
Qed.
-Lemma for_all_filter
+Lemma for_all_filter:
forall s, for_all f s=is_empty (filter (fun x => negb (f x)) s).
Proof.
intros; apply bool_1; split; intros.
@@ -638,7 +638,7 @@ rewrite filter_iff; auto.
destruct (f x); auto.
Qed.
-Lemma exists_filter
+Lemma exists_filter :
forall s, exists_ f s=negb (is_empty (filter f s)).
Proof.
intros; apply bool_1; split; intros.
@@ -653,19 +653,19 @@ intros _ H0.
rewrite (is_empty_1 (H0 (refl_equal None))) in H; auto; discriminate.
Qed.
-Lemma partition_filter_1
+Lemma partition_filter_1:
forall s, equal (fst (partition f s)) (filter f s)=true.
Proof.
auto.
Qed.
-Lemma partition_filter_2
+Lemma partition_filter_2:
forall s, equal (snd (partition f s)) (filter (fun x => negb (f x)) s)=true.
Proof.
auto.
Qed.
-Lemma add_filter_1 forall s s' x,
+Lemma add_filter_1 : forall s s' x,
f x=true -> (Add x s s') -> (Add x (filter f s) (filter f s')).
Proof.
unfold Add, MP.Add; intros.
@@ -676,7 +676,7 @@ assert (E.eq x y -> f y = true) by
tauto.
Qed.
-Lemma add_filter_2 forall s s' x,
+Lemma add_filter_2 : forall s s' x,
f x=false -> (Add x s s') -> filter f s [=] filter f s'.
Proof.
unfold Add, MP.Add, Equal; intros.
@@ -689,7 +689,7 @@ assert (f a = true -> ~E.eq x a).
tauto.
Qed.
-Lemma union_filter forall f g, (compat_bool E.eq f) -> (compat_bool E.eq g) ->
+Lemma union_filter: forall f g, (compat_bool E.eq f) -> (compat_bool E.eq g) ->
forall s, union (filter f s) (filter g s) [=] filter (fun x=>orb (f x) (g x)) s.
Proof.
clear Comp' Comp f.
@@ -706,7 +706,7 @@ Qed.
(** Properties of [for_all] *)
-Lemma for_all_mem_1 forall s,
+Lemma for_all_mem_1: forall s,
(forall x, (mem x s)=true->(f x)=true) -> (for_all f s)=true.
Proof.
intros.
@@ -719,7 +719,7 @@ generalize (H a); case (mem a s);intros;auto.
rewrite H0;auto.
Qed.
-Lemma for_all_mem_2 forall s,
+Lemma for_all_mem_2: forall s,
(for_all f s)=true -> forall x,(mem x s)=true -> (f x)=true.
Proof.
intros.
@@ -732,7 +732,7 @@ rewrite H0; simpl;intros.
replace true with (negb false);auto;apply negb_sym;auto.
Qed.
-Lemma for_all_mem_3
+Lemma for_all_mem_3:
forall s x,(mem x s)=true -> (f x)=false -> (for_all f s)=false.
Proof.
intros.
@@ -747,8 +747,8 @@ rewrite H0.
simpl;auto.
Qed.
-Lemma for_all_mem_4
- forall s, for_all f s=false -> {xelt | mem x s=true /\ f x=false}.
+Lemma for_all_mem_4:
+ forall s, for_all f s=false -> {x:elt | mem x s=true /\ f x=false}.
Proof.
intros.
rewrite for_all_filter in H; auto.
@@ -762,7 +762,7 @@ Qed.
(** Properties of [exists] *)
-Lemma for_all_exists
+Lemma for_all_exists:
forall s, exists_ f s = negb (for_all (fun x =>negb (f x)) s).
Proof.
intros.
@@ -775,15 +775,15 @@ Qed.
End Bool.
Section Bool'.
-Variable felt->bool.
-Variable Comp compat_bool E.eq f.
+Variable f:elt->bool.
+Variable Comp: compat_bool E.eq f.
-Let Comp' compat_bool E.eq (fun x =>negb (f x)).
+Let Comp' : compat_bool E.eq (fun x =>negb (f x)).
Proof.
unfold compat_bool in *; intros; f_equal; auto.
Qed.
-Lemma exists_mem_1
+Lemma exists_mem_1:
forall s, (forall x, mem x s=true->f x=false) -> exists_ f s=false.
Proof.
intros.
@@ -793,7 +793,7 @@ intros;generalize (H x H0);intros.
symmetry;apply negb_sym;simpl;auto.
Qed.
-Lemma exists_mem_2
+Lemma exists_mem_2:
forall s, exists_ f s=false -> forall x, mem x s=true -> f x=false.
Proof.
intros.
@@ -803,7 +803,7 @@ rewrite (for_all_mem_2 (fun x => negb (f x)) Comp' s);simpl;auto.
replace true with (negb false);auto;apply negb_sym;auto.
Qed.
-Lemma exists_mem_3
+Lemma exists_mem_3:
forall s x, mem x s=true -> f x=true -> exists_ f s=true.
Proof.
intros.
@@ -813,8 +813,8 @@ apply for_all_mem_3 with x;auto.
rewrite H0;auto.
Qed.
-Lemma exists_mem_4
- forall s, exists_ f s=true -> {xelt | (mem x s)=true /\ (f x)=true}.
+Lemma exists_mem_4:
+ forall s, exists_ f s=true -> {x:elt | (mem x s)=true /\ (f x)=true}.
Proof.
intros.
rewrite for_all_exists in H; auto.
@@ -831,21 +831,21 @@ Section Sum.
(** Adding a valuation function on all elements of a set. *)
-Definition sum (felt -> nat)(s:t) := fold (fun x => plus (f x)) s 0.
+Definition sum (f:elt -> nat)(s:t) := fold (fun x => plus (f x)) s 0.
-Lemma sum_plus
+Lemma sum_plus :
forall f g, compat_nat E.eq f -> compat_nat E.eq g ->
forall s, sum (fun x =>f x+g x) s = sum f s + sum g s.
Proof.
unfold sum.
intros f g Hf Hg.
-assert (fc compat_op E.eq (@eq _) (fun x:elt =>plus (f x))). auto.
-assert (ft transpose (@eq _) (fun x:elt =>plus (f x))). red; intros; omega.
-assert (gc compat_op E.eq (@eq _) (fun x:elt => plus (g x))). auto.
-assert (gt transpose (@eq _) (fun x:elt =>plus (g x))). red; intros; omega.
-assert (fgc compat_op E.eq (@eq _) (fun x:elt =>plus ((f x)+(g x)))). auto.
-assert (fgt transpose (@eq _) (fun x:elt=>plus ((f x)+(g x)))). red; intros; omega.
-assert (st = gen_st nat).
+assert (fc : compat_op E.eq (@eq _) (fun x:elt =>plus (f x))). auto.
+assert (ft : transpose (@eq _) (fun x:elt =>plus (f x))). red; intros; omega.
+assert (gc : compat_op E.eq (@eq _) (fun x:elt => plus (g x))). auto.
+assert (gt : transpose (@eq _) (fun x:elt =>plus (g x))). red; intros; omega.
+assert (fgc : compat_op E.eq (@eq _) (fun x:elt =>plus ((f x)+(g x)))). auto.
+assert (fgt : transpose (@eq _) (fun x:elt=>plus ((f x)+(g x)))). red; intros; omega.
+assert (st := gen_st nat).
intros s;pattern s; apply set_rec.
intros.
rewrite <- (fold_equal _ _ st _ fc ft 0 _ _ H).
@@ -856,15 +856,15 @@ rewrite H0;simpl;omega.
intros; do 3 rewrite (fold_empty _ _ st);auto.
Qed.
-Lemma sum_filter forall f, (compat_bool E.eq f) ->
+Lemma sum_filter : forall f, (compat_bool E.eq f) ->
forall s, (sum (fun x => if f x then 1 else 0) s) = (cardinal (filter f s)).
Proof.
unfold sum; intros f Hf.
-assert (st = gen_st nat).
-assert (cc compat_op E.eq (@eq _) (fun x => plus (if f x then 1 else 0))).
+assert (st := gen_st nat).
+assert (cc : compat_op E.eq (@eq _) (fun x => plus (if f x then 1 else 0))).
unfold compat_op; intros.
rewrite (Hf x x' H); auto.
-assert (ct transpose (@eq _) (fun x => plus (if f x then 1 else 0))).
+assert (ct : transpose (@eq _) (fun x => plus (if f x then 1 else 0))).
unfold transpose; intros; omega.
intros s;pattern s; apply set_rec.
intros.
@@ -884,32 +884,32 @@ unfold Empty; intros.
rewrite filter_iff; auto; set_iff; tauto.
Qed.
-Lemma fold_compat
- forall (ASet)(eqA:A->A->Prop)(st:(Setoid_Theory _ eqA))
- (f gelt->A->A),
+Lemma fold_compat :
+ forall (A:Set)(eqA:A->A->Prop)(st:(Setoid_Theory _ eqA))
+ (f g:elt->A->A),
(compat_op E.eq eqA f) -> (transpose eqA f) ->
(compat_op E.eq eqA g) -> (transpose eqA g) ->
- forall (iA)(s:t),(forall x:elt, (In x s) -> forall y, (eqA (f x y) (g x y))) ->
+ forall (i:A)(s:t),(forall x:elt, (In x s) -> forall y, (eqA (f x y) (g x y))) ->
(eqA (fold f s i) (fold g s i)).
Proof.
intros A eqA st f g fc ft gc gt i.
intro s; pattern s; apply set_rec; intros.
trans_st (fold f s0 i).
-apply fold_equal with (eqA=eqA); auto.
+apply fold_equal with (eqA:=eqA); auto.
rewrite equal_sym; auto.
trans_st (fold g s0 i).
apply H0; intros; apply H1; auto.
elim (equal_2 H x); auto; intros.
-apply fold_equal with (eqA=eqA); auto.
+apply fold_equal with (eqA:=eqA); auto.
trans_st (f x (fold f s0 i)).
-apply fold_add with (eqA=eqA); auto.
+apply fold_add with (eqA:=eqA); auto.
trans_st (g x (fold f s0 i)).
trans_st (g x (fold g s0 i)).
-sym_st; apply fold_add with (eqA=eqA); auto.
+sym_st; apply fold_add with (eqA:=eqA); auto.
trans_st i; [idtac | sym_st ]; apply fold_empty; auto.
Qed.
-Lemma sum_compat
+Lemma sum_compat :
forall f g, compat_nat E.eq f -> compat_nat E.eq g ->
forall s, (forall x, In x s -> f x=g x) -> sum f s=sum g s.
intros.
diff --git a/theories/FSets/FSetFacts.v b/theories/FSets/FSetFacts.v
index b7756592f..5cf2808c5 100644
--- a/theories/FSets/FSetFacts.v
+++ b/theories/FSets/FSetFacts.v
@@ -6,13 +6,13 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetFacts.v,v 1.6 2006/03/13 04:59:24 letouzey Exp $ *)
(** * Finite sets library *)
(** This functor derives additional facts from [FSetInterface.S]. These
facts are mainly the specifications of [FSetInterface.S] written using
- different styles equivalence and boolean equalities.
+ different styles: equivalence and boolean equalities.
Moreover, we prove that [E.Eq] and [Equal] are setoid equalities.
*)
@@ -20,8 +20,8 @@ Require Export FSetInterface.
Set Implicit Arguments.
Unset Strict Implicit.
-Module Facts (M S).
-Module ME = OrderedTypeFacts M.E.
+Module Facts (M: S).
+Module ME := OrderedTypeFacts M.E.
Import ME.
Import M.
Import Logic. (* to unmask [eq] *)
@@ -30,108 +30,108 @@ Import Peano. (* to unmask [lt] *)
(** * Specifications written using equivalences *)
Section IffSpec.
-Variable s s' s'' t.
-Variable x y z elt.
+Variable s s' s'' : t.
+Variable x y z : elt.
-Lemma In_eq_iff E.eq x y -> (In x s <-> In y s).
+Lemma In_eq_iff : E.eq x y -> (In x s <-> In y s).
Proof.
split; apply In_1; auto.
Qed.
-Lemma mem_iff In x s <-> mem x s = true.
+Lemma mem_iff : In x s <-> mem x s = true.
Proof.
split; [apply mem_1|apply mem_2].
Qed.
-Lemma not_mem_iff ~In x s <-> mem x s = false.
+Lemma not_mem_iff : ~In x s <-> mem x s = false.
Proof.
rewrite mem_iff; destruct (mem x s); intuition.
Qed.
-Lemma equal_iff s[=]s' <-> equal s s' = true.
+Lemma equal_iff : s[=]s' <-> equal s s' = true.
Proof.
split; [apply equal_1|apply equal_2].
Qed.
-Lemma subset_iff s[<=]s' <-> subset s s' = true.
+Lemma subset_iff : s[<=]s' <-> subset s s' = true.
Proof.
split; [apply subset_1|apply subset_2].
Qed.
-Lemma empty_iff In x empty <-> False.
+Lemma empty_iff : In x empty <-> False.
Proof.
intuition; apply (empty_1 H).
Qed.
-Lemma is_empty_iff Empty s <-> is_empty s = true.
+Lemma is_empty_iff : Empty s <-> is_empty s = true.
Proof.
split; [apply is_empty_1|apply is_empty_2].
Qed.
-Lemma singleton_iff In y (singleton x) <-> E.eq x y.
+Lemma singleton_iff : In y (singleton x) <-> E.eq x y.
Proof.
split; [apply singleton_1|apply singleton_2].
Qed.
-Lemma add_iff In y (add x s) <-> E.eq x y \/ In y s.
+Lemma add_iff : In y (add x s) <-> E.eq x y \/ In y s.
Proof.
split; [ | destruct 1; [apply add_1|apply add_2]]; auto.
destruct (eq_dec x y) as [E|E]; auto.
intro H; right; exact (add_3 E H).
Qed.
-Lemma add_neq_iff ~ E.eq x y -> (In y (add x s) <-> In y s).
+Lemma add_neq_iff : ~ E.eq x y -> (In y (add x s) <-> In y s).
Proof.
split; [apply add_3|apply add_2]; auto.
Qed.
-Lemma remove_iff In y (remove x s) <-> In y s /\ ~E.eq x y.
+Lemma remove_iff : In y (remove x s) <-> In y s /\ ~E.eq x y.
Proof.
split; [split; [apply remove_3 with x |] | destruct 1; apply remove_2]; auto.
intro.
apply (remove_1 H0 H).
Qed.
-Lemma remove_neq_iff ~ E.eq x y -> (In y (remove x s) <-> In y s).
+Lemma remove_neq_iff : ~ E.eq x y -> (In y (remove x s) <-> In y s).
Proof.
split; [apply remove_3|apply remove_2]; auto.
Qed.
-Lemma union_iff In x (union s s') <-> In x s \/ In x s'.
+Lemma union_iff : In x (union s s') <-> In x s \/ In x s'.
Proof.
split; [apply union_1 | destruct 1; [apply union_2|apply union_3]]; auto.
Qed.
-Lemma inter_iff In x (inter s s') <-> In x s /\ In x s'.
+Lemma inter_iff : In x (inter s s') <-> In x s /\ In x s'.
Proof.
split; [split; [apply inter_1 with s' | apply inter_2 with s] | destruct 1; apply inter_3]; auto.
Qed.
-Lemma diff_iff In x (diff s s') <-> In x s /\ ~ In x s'.
+Lemma diff_iff : In x (diff s s') <-> In x s /\ ~ In x s'.
Proof.
split; [split; [apply diff_1 with s' | apply diff_2 with s] | destruct 1; apply diff_3]; auto.
Qed.
-Variable f elt->bool.
+Variable f : elt->bool.
-Lemma filter_iff compat_bool E.eq f -> (In x (filter f s) <-> In x s /\ f x = true).
+Lemma filter_iff : compat_bool E.eq f -> (In x (filter f s) <-> In x s /\ f x = true).
Proof.
split; [split; [apply filter_1 with f | apply filter_2 with s] | destruct 1; apply filter_3]; auto.
Qed.
-Lemma for_all_iff compat_bool E.eq f ->
+Lemma for_all_iff : compat_bool E.eq f ->
(For_all (fun x => f x = true) s <-> for_all f s = true).
Proof.
split; [apply for_all_1 | apply for_all_2]; auto.
Qed.
-Lemma exists_iff compat_bool E.eq f ->
+Lemma exists_iff : compat_bool E.eq f ->
(Exists (fun x => f x = true) s <-> exists_ f s = true).
Proof.
split; [apply exists_1 | apply exists_2]; auto.
Qed.
-Lemma elements_iff In x s <-> ME.In x (elements s).
+Lemma elements_iff : In x s <-> ME.In x (elements s).
Proof.
split; [apply elements_1 | apply elements_2].
Qed.
@@ -140,7 +140,7 @@ End IffSpec.
(** Useful tactic for simplifying expressions like [In y (add x (union s s'))] *)
-Ltac set_iff =
+Ltac set_iff :=
repeat (progress (
rewrite add_iff || rewrite remove_iff || rewrite singleton_iff
|| rewrite union_iff || rewrite inter_iff || rewrite diff_iff
@@ -149,65 +149,65 @@ Ltac set_iff =
(** * Specifications written using boolean predicates *)
Section BoolSpec.
-Variable s s' s'' t.
-Variable x y z elt.
+Variable s s' s'' : t.
+Variable x y z : elt.
-Lemma mem_b E.eq x y -> mem x s = mem y s.
+Lemma mem_b : E.eq x y -> mem x s = mem y s.
Proof.
intros.
generalize (mem_iff s x) (mem_iff s y)(In_eq_iff s H).
destruct (mem x s); destruct (mem y s); intuition.
Qed.
-Lemma add_b mem y (add x s) = eqb x y || mem y s.
+Lemma add_b : mem y (add x s) = eqb x y || mem y s.
Proof.
generalize (mem_iff (add x s) y)(mem_iff s y)(add_iff s x y); unfold eqb.
destruct (eq_dec x y); destruct (mem y s); destruct (mem y (add x s)); intuition.
Qed.
-Lemma add_neq_b ~ E.eq x y -> mem y (add x s) = mem y s.
+Lemma add_neq_b : ~ E.eq x y -> mem y (add x s) = mem y s.
Proof.
intros; generalize (mem_iff (add x s) y)(mem_iff s y)(add_neq_iff s H).
destruct (mem y s); destruct (mem y (add x s)); intuition.
Qed.
-Lemma remove_b mem y (remove x s) = mem y s && negb (eqb x y).
+Lemma remove_b : mem y (remove x s) = mem y s && negb (eqb x y).
Proof.
generalize (mem_iff (remove x s) y)(mem_iff s y)(remove_iff s x y); unfold eqb.
destruct (eq_dec x y); destruct (mem y s); destruct (mem y (remove x s)); simpl; intuition.
Qed.
-Lemma remove_neq_b ~ E.eq x y -> mem y (remove x s) = mem y s.
+Lemma remove_neq_b : ~ E.eq x y -> mem y (remove x s) = mem y s.
Proof.
intros; generalize (mem_iff (remove x s) y)(mem_iff s y)(remove_neq_iff s H).
destruct (mem y s); destruct (mem y (remove x s)); intuition.
Qed.
-Lemma singleton_b mem y (singleton x) = eqb x y.
+Lemma singleton_b : mem y (singleton x) = eqb x y.
Proof.
generalize (mem_iff (singleton x) y)(singleton_iff x y); unfold eqb.
destruct (eq_dec x y); destruct (mem y (singleton x)); intuition.
Qed.
-Lemma union_b mem x (union s s') = mem x s || mem x s'.
+Lemma union_b : mem x (union s s') = mem x s || mem x s'.
Proof.
generalize (mem_iff (union s s') x)(mem_iff s x)(mem_iff s' x)(union_iff s s' x).
destruct (mem x s); destruct (mem x s'); destruct (mem x (union s s')); intuition.
Qed.
-Lemma inter_b mem x (inter s s') = mem x s && mem x s'.
+Lemma inter_b : mem x (inter s s') = mem x s && mem x s'.
Proof.
generalize (mem_iff (inter s s') x)(mem_iff s x)(mem_iff s' x)(inter_iff s s' x).
destruct (mem x s); destruct (mem x s'); destruct (mem x (inter s s')); intuition.
Qed.
-Lemma diff_b mem x (diff s s') = mem x s && negb (mem x s').
+Lemma diff_b : mem x (diff s s') = mem x s && negb (mem x s').
Proof.
generalize (mem_iff (diff s s') x)(mem_iff s x)(mem_iff s' x)(diff_iff s s' x).
destruct (mem x s); destruct (mem x s'); destruct (mem x (diff s s')); simpl; intuition.
Qed.
-Lemma elements_b mem x s = existsb (eqb x) (elements s).
+Lemma elements_b : mem x s = existsb (eqb x) (elements s).
Proof.
generalize (mem_iff s x)(elements_iff s x)(existsb_exists (eqb x) (elements s)).
rewrite InA_alt.
@@ -226,16 +226,16 @@ exists a; intuition.
unfold eqb in *; destruct (eq_dec x a); auto; discriminate.
Qed.
-Variable f elt->bool.
+Variable f : elt->bool.
-Lemma filter_b compat_bool E.eq f -> mem x (filter f s) = mem x s && f x.
+Lemma filter_b : compat_bool E.eq f -> mem x (filter f s) = mem x s && f x.
Proof.
intros.
generalize (mem_iff (filter f s) x)(mem_iff s x)(filter_iff s x H).
destruct (mem x s); destruct (mem x (filter f s)); destruct (f x); simpl; intuition.
Qed.
-Lemma for_all_b compat_bool E.eq f ->
+Lemma for_all_b : compat_bool E.eq f ->
for_all f s = forallb f (elements s).
Proof.
intros.
@@ -255,7 +255,7 @@ destruct H1 as (_,H1).
apply H1; auto.
Qed.
-Lemma exists_b compat_bool E.eq f ->
+Lemma exists_b : compat_bool E.eq f ->
exists_ f s = existsb f (elements s).
Proof.
intros.
@@ -281,27 +281,27 @@ End BoolSpec.
(** * [E.eq] and [Equal] are setoid equalities *)
-Definition E_ST Setoid_Theory elt E.eq.
+Definition E_ST : Setoid_Theory elt E.eq.
Proof.
constructor; [apply E.eq_refl|apply E.eq_sym|apply E.eq_trans].
Qed.
Add Setoid elt E.eq E_ST as EltSetoid.
-Definition Equal_ST Setoid_Theory t Equal.
+Definition Equal_ST : Setoid_Theory t Equal.
Proof.
constructor; [apply eq_refl | apply eq_sym | apply eq_trans].
Qed.
Add Setoid t Equal Equal_ST as EqualSetoid.
-Add Morphism In In_m.
+Add Morphism In : In_m.
Proof.
unfold Equal; intros x y H s s' H0.
rewrite (In_eq_iff s H); auto.
Qed.
-Add Morphism is_empty is_empty_m.
+Add Morphism is_empty : is_empty_m.
Proof.
unfold Equal; intros s s' H.
generalize (is_empty_iff s)(is_empty_iff s').
@@ -318,12 +318,12 @@ destruct H1 as (_,H1).
exact (H1 (refl_equal true) _ Ha).
Qed.
-Add Morphism Empty Empty_m.
+Add Morphism Empty : Empty_m.
Proof.
intros; do 2 rewrite is_empty_iff; rewrite H; intuition.
Qed.
-Add Morphism mem mem_m.
+Add Morphism mem : mem_m.
Proof.
unfold Equal; intros x y H s s' H0.
generalize (H0 x); clear H0; rewrite (In_eq_iff s' H).
@@ -331,48 +331,48 @@ generalize (mem_iff s x)(mem_iff s' y).
destruct (mem x s); destruct (mem y s'); intuition.
Qed.
-Add Morphism singleton singleton_m.
+Add Morphism singleton : singleton_m.
Proof.
unfold Equal; intros x y H a.
do 2 rewrite singleton_iff; split; order.
Qed.
-Add Morphism add add_m.
+Add Morphism add : add_m.
Proof.
unfold Equal; intros x y H s s' H0 a.
do 2 rewrite add_iff; rewrite H; rewrite H0; intuition.
Qed.
-Add Morphism remove remove_m.
+Add Morphism remove : remove_m.
Proof.
unfold Equal; intros x y H s s' H0 a.
do 2 rewrite remove_iff; rewrite H; rewrite H0; intuition.
Qed.
-Add Morphism union union_m.
+Add Morphism union : union_m.
Proof.
unfold Equal; intros s s' H s'' s''' H0 a.
do 2 rewrite union_iff; rewrite H; rewrite H0; intuition.
Qed.
-Add Morphism inter inter_m.
+Add Morphism inter : inter_m.
Proof.
unfold Equal; intros s s' H s'' s''' H0 a.
do 2 rewrite inter_iff; rewrite H; rewrite H0; intuition.
Qed.
-Add Morphism diff diff_m.
+Add Morphism diff : diff_m.
Proof.
unfold Equal; intros s s' H s'' s''' H0 a.
do 2 rewrite diff_iff; rewrite H; rewrite H0; intuition.
Qed.
-Add Morphism Subset Subset_m.
+Add Morphism Subset : Subset_m.
Proof.
unfold Equal, Subset; firstorder.
Qed.
-Add Morphism subset subset_m.
+Add Morphism subset : subset_m.
Proof.
intros s s' H s'' s''' H0.
generalize (subset_iff s s'') (subset_iff s' s''').
@@ -381,7 +381,7 @@ rewrite H in H1; rewrite H0 in H1; intuition.
rewrite H in H1; rewrite H0 in H1; intuition.
Qed.
-Add Morphism equal equal_m.
+Add Morphism equal : equal_m.
Proof.
intros s s' H s'' s''' H0.
generalize (equal_iff s s'') (equal_iff s' s''').
@@ -391,9 +391,9 @@ rewrite H in H1; rewrite H0 in H1; intuition.
Qed.
(* [fold], [filter], [for_all], [exists_] and [partition] cannot be proved morphism
- without additional hypothesis on [f]. For instance *)
+ without additional hypothesis on [f]. For instance: *)
-Lemma filter_equal forall f, compat_bool E.eq f ->
+Lemma filter_equal : forall f, compat_bool E.eq f ->
forall s s', s[=]s' -> filter f s [=] filter f s'.
Proof.
unfold Equal; intros; repeat rewrite filter_iff; auto; rewrite H0; tauto.
@@ -402,7 +402,7 @@ Qed.
(* For [elements], [min_elt], [max_elt] and [choose], we would need setoid
structures on [list elt] and [option elt]. *)
-(* Later
+(* Later:
Add Morphism cardinal ; cardinal_m.
*)
diff --git a/theories/FSets/FSetInterface.v b/theories/FSets/FSetInterface.v
index ad37e0db6..ff3d7e578 100644
--- a/theories/FSets/FSetInterface.v
+++ b/theories/FSets/FSetInterface.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetInterface.v,v 1.11 2006/03/10 10:49:48 letouzey Exp $ *)
(** * Finite set library *)
@@ -18,12 +18,12 @@ Set Implicit Arguments.
Unset Strict Implicit.
(** Compatibility of a boolean function with respect to an equality. *)
-Definition compat_bool (ASet)(eqA: A->A->Prop)(f: A-> bool) :=
- forall x y A, eqA x y -> f x = f y.
+Definition compat_bool (A:Set)(eqA: A->A->Prop)(f: A-> bool) :=
+ forall x y : A, eqA x y -> f x = f y.
(** Compatibility of a predicate with respect to an equality. *)
-Definition compat_P (ASet)(eqA: A->A->Prop)(P : A -> Prop) :=
- forall x y A, eqA x y -> P x -> P y.
+Definition compat_P (A:Set)(eqA: A->A->Prop)(P : A -> Prop) :=
+ forall x y : A, eqA x y -> P x -> P y.
Hint Unfold compat_bool compat_P.
@@ -34,243 +34,243 @@ Hint Unfold compat_bool compat_P.
Module Type S.
- Declare Module E OrderedType.
- Definition elt = E.t.
+ Declare Module E : OrderedType.
+ Definition elt := E.t.
- Parameter t Set. (** the abstract type of sets *)
+ Parameter t : Set. (** the abstract type of sets *)
(** Logical predicates *)
- Parameter In elt -> t -> Prop.
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) s := forall x, In x s -> P x.
- Definition Exists (P elt -> Prop) s := exists x, In x s /\ P x.
+ Parameter In : elt -> t -> Prop.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
+ Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.
- Notation "s [=] t" = (Equal s t) (at level 70, no associativity).
- Notation "s [<=] t" = (Subset s t) (at level 70, no associativity).
+ Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
+ Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).
- Parameter empty t.
+ Parameter empty : t.
(** The empty set. *)
- Parameter is_empty t -> bool.
+ Parameter is_empty : t -> bool.
(** Test whether a set is empty or not. *)
- Parameter mem elt -> t -> bool.
+ Parameter mem : elt -> t -> bool.
(** [mem x s] tests whether [x] belongs to the set [s]. *)
- Parameter add elt -> t -> t.
+ Parameter add : elt -> t -> t.
(** [add x s] returns a set containing all elements of [s],
plus [x]. If [x] was already in [s], [s] is returned unchanged. *)
- Parameter singleton elt -> t.
+ Parameter singleton : elt -> t.
(** [singleton x] returns the one-element set containing only [x]. *)
- Parameter remove elt -> t -> t.
+ Parameter remove : elt -> t -> t.
(** [remove x s] returns a set containing all elements of [s],
except [x]. If [x] was not in [s], [s] is returned unchanged. *)
- Parameter union t -> t -> t.
+ Parameter union : t -> t -> t.
(** Set union. *)
- Parameter inter t -> t -> t.
+ Parameter inter : t -> t -> t.
(** Set intersection. *)
- Parameter diff t -> t -> t.
+ Parameter diff : t -> t -> t.
(** Set difference. *)
- Definition eq t -> t -> Prop := Equal.
- Parameter lt t -> t -> Prop.
- Parameter compare forall s s' : t, Compare lt eq s s'.
+ Definition eq : t -> t -> Prop := Equal.
+ Parameter lt : t -> t -> Prop.
+ Parameter compare : forall s s' : t, Compare lt eq s s'.
(** Total ordering between sets. Can be used as the ordering function
for doing sets of sets. *)
- Parameter equal t -> t -> bool.
+ Parameter equal : t -> t -> bool.
(** [equal s1 s2] tests whether the sets [s1] and [s2] are
equal, that is, contain equal elements. *)
- Parameter subset t -> t -> bool.
+ Parameter subset : t -> t -> bool.
(** [subset s1 s2] tests whether the set [s1] is a subset of
the set [s2]. *)
- (** Coq comment [iter] is useless in a purely functional world *)
- (** iter (elt -> unit) -> set -> unit. i*)
+ (** Coq comment: [iter] is useless in a purely functional world *)
+ (** iter: (elt -> unit) -> set -> unit. i*)
(** [iter f s] applies [f] in turn to all elements of [s].
The order in which the elements of [s] are presented to [f]
is unspecified. *)
- Parameter fold forall A : Set, (elt -> A -> A) -> t -> A -> A.
+ Parameter fold : forall A : Set, (elt -> A -> A) -> t -> A -> A.
(** [fold f s a] computes [(f xN ... (f x2 (f x1 a))...)],
where [x1 ... xN] are the elements of [s], in increasing order. *)
- Parameter for_all (elt -> bool) -> t -> bool.
+ Parameter for_all : (elt -> bool) -> t -> bool.
(** [for_all p s] checks if all elements of the set
satisfy the predicate [p]. *)
- Parameter exists_ (elt -> bool) -> t -> bool.
+ Parameter exists_ : (elt -> bool) -> t -> bool.
(** [exists p s] checks if at least one element of
the set satisfies the predicate [p]. *)
- Parameter filter (elt -> bool) -> t -> t.
+ Parameter filter : (elt -> bool) -> t -> t.
(** [filter p s] returns the set of all elements in [s]
that satisfy predicate [p]. *)
- Parameter partition (elt -> bool) -> t -> t * t.
+ Parameter partition : (elt -> bool) -> t -> t * t.
(** [partition p s] returns a pair of sets [(s1, s2)], where
[s1] is the set of all the elements of [s] that satisfy the
predicate [p], and [s2] is the set of all the elements of
[s] that do not satisfy [p]. *)
- Parameter cardinal t -> nat.
+ Parameter cardinal : t -> nat.
(** Return the number of elements of a set. *)
- (** Coq comment nat instead of int ... *)
+ (** Coq comment: nat instead of int ... *)
- Parameter elements t -> list elt.
+ Parameter elements : t -> list elt.
(** Return the list of all elements of the given set.
The returned list is sorted in increasing order with respect
to the ordering [Ord.compare], where [Ord] is the argument
given to {!Set.Make}. *)
- Parameter min_elt t -> option elt.
+ Parameter min_elt : t -> option elt.
(** Return the smallest element of the given set
(with respect to the [Ord.compare] ordering), or raise
[Not_found] if the set is empty. *)
- (** Coq comment [Not_found] is represented by the option type *)
+ (** Coq comment: [Not_found] is represented by the option type *)
- Parameter max_elt t -> option elt.
+ Parameter max_elt : t -> option elt.
(** Same as {!Set.S.min_elt}, but returns the largest element of the
given set. *)
- (** Coq comment [Not_found] is represented by the option type *)
+ (** Coq comment: [Not_found] is represented by the option type *)
- Parameter choose t -> option elt.
+ Parameter choose : t -> option elt.
(** Return one element of the given set, or raise [Not_found] if
the set is empty. Which element is chosen is unspecified,
but equal elements will be chosen for equal sets. *)
- (** Coq comment [Not_found] is represented by the option type *)
+ (** Coq comment: [Not_found] is represented by the option type *)
Section Spec.
- Variable s s' s'' t.
- Variable x y z elt.
+ Variable s s' s'' : t.
+ Variable x y z : elt.
(** Specification of [In] *)
- Parameter In_1 E.eq x y -> In x s -> In y s.
+ Parameter In_1 : E.eq x y -> In x s -> In y s.
(** Specification of [eq] *)
- Parameter eq_refl eq s s.
- Parameter eq_sym eq s s' -> eq s' s.
- Parameter eq_trans eq s s' -> eq s' s'' -> eq s s''.
+ Parameter eq_refl : eq s s.
+ Parameter eq_sym : eq s s' -> eq s' s.
+ Parameter eq_trans : eq s s' -> eq s' s'' -> eq s s''.
(** Specification of [lt] *)
- Parameter lt_trans lt s s' -> lt s' s'' -> lt s s''.
- Parameter lt_not_eq lt s s' -> ~ eq s s'.
+ Parameter lt_trans : lt s s' -> lt s' s'' -> lt s s''.
+ Parameter lt_not_eq : lt s s' -> ~ eq s s'.
(** Specification of [mem] *)
- Parameter mem_1 In x s -> mem x s = true.
- Parameter mem_2 mem x s = true -> In x s.
+ Parameter mem_1 : In x s -> mem x s = true.
+ Parameter mem_2 : mem x s = true -> In x s.
(** Specification of [equal] *)
- Parameter equal_1 s[=]s' -> equal s s' = true.
- Parameter equal_2 equal s s' = true ->s[=]s'.
+ Parameter equal_1 : s[=]s' -> equal s s' = true.
+ Parameter equal_2 : equal s s' = true ->s[=]s'.
(** Specification of [subset] *)
- Parameter subset_1 s[<=]s' -> subset s s' = true.
- Parameter subset_2 subset s s' = true -> s[<=]s'.
+ Parameter subset_1 : s[<=]s' -> subset s s' = true.
+ Parameter subset_2 : subset s s' = true -> s[<=]s'.
(** Specification of [empty] *)
- Parameter empty_1 Empty empty.
+ Parameter empty_1 : Empty empty.
(** Specification of [is_empty] *)
- Parameter is_empty_1 Empty s -> is_empty s = true.
- Parameter is_empty_2 is_empty s = true -> Empty s.
+ Parameter is_empty_1 : Empty s -> is_empty s = true.
+ Parameter is_empty_2 : is_empty s = true -> Empty s.
(** Specification of [add] *)
- Parameter add_1 E.eq x y -> In y (add x s).
- Parameter add_2 In y s -> In y (add x s).
- Parameter add_3 ~ E.eq x y -> In y (add x s) -> In y s.
+ Parameter add_1 : E.eq x y -> In y (add x s).
+ Parameter add_2 : In y s -> In y (add x s).
+ Parameter add_3 : ~ E.eq x y -> In y (add x s) -> In y s.
(** Specification of [remove] *)
- Parameter remove_1 E.eq x y -> ~ In y (remove x s).
- Parameter remove_2 ~ E.eq x y -> In y s -> In y (remove x s).
- Parameter remove_3 In y (remove x s) -> In y s.
+ Parameter remove_1 : E.eq x y -> ~ In y (remove x s).
+ Parameter remove_2 : ~ E.eq x y -> In y s -> In y (remove x s).
+ Parameter remove_3 : In y (remove x s) -> In y s.
(** Specification of [singleton] *)
- Parameter singleton_1 In y (singleton x) -> E.eq x y.
- Parameter singleton_2 E.eq x y -> In y (singleton x).
+ Parameter singleton_1 : In y (singleton x) -> E.eq x y.
+ Parameter singleton_2 : E.eq x y -> In y (singleton x).
(** Specification of [union] *)
- Parameter union_1 In x (union s s') -> In x s \/ In x s'.
- Parameter union_2 In x s -> In x (union s s').
- Parameter union_3 In x s' -> In x (union s s').
+ Parameter union_1 : In x (union s s') -> In x s \/ In x s'.
+ Parameter union_2 : In x s -> In x (union s s').
+ Parameter union_3 : In x s' -> In x (union s s').
(** Specification of [inter] *)
- Parameter inter_1 In x (inter s s') -> In x s.
- Parameter inter_2 In x (inter s s') -> In x s'.
- Parameter inter_3 In x s -> In x s' -> In x (inter s s').
+ Parameter inter_1 : In x (inter s s') -> In x s.
+ Parameter inter_2 : In x (inter s s') -> In x s'.
+ Parameter inter_3 : In x s -> In x s' -> In x (inter s s').
(** Specification of [diff] *)
- Parameter diff_1 In x (diff s s') -> In x s.
- Parameter diff_2 In x (diff s s') -> ~ In x s'.
- Parameter diff_3 In x s -> ~ In x s' -> In x (diff s s').
+ Parameter diff_1 : In x (diff s s') -> In x s.
+ Parameter diff_2 : In x (diff s s') -> ~ In x s'.
+ Parameter diff_3 : In x s -> ~ In x s' -> In x (diff s s').
(** Specification of [fold] *)
- Parameter fold_1 forall (A : Set) (i : A) (f : elt -> A -> A),
+ Parameter fold_1 : forall (A : Set) (i : A) (f : elt -> A -> A),
fold f s i = fold_left (fun a e => f e a) (elements s) i.
(** Specification of [cardinal] *)
- Parameter cardinal_1 cardinal s = length (elements s).
+ Parameter cardinal_1 : cardinal s = length (elements s).
Section Filter.
- Variable f elt -> bool.
+ Variable f : elt -> bool.
(** Specification of [filter] *)
- Parameter filter_1 compat_bool E.eq f -> In x (filter f s) -> In x s.
- Parameter filter_2 compat_bool E.eq f -> In x (filter f s) -> f x = true.
- Parameter filter_3
+ Parameter filter_1 : compat_bool E.eq f -> In x (filter f s) -> In x s.
+ Parameter filter_2 : compat_bool E.eq f -> In x (filter f s) -> f x = true.
+ Parameter filter_3 :
compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s).
(** Specification of [for_all] *)
- Parameter for_all_1
+ Parameter for_all_1 :
compat_bool E.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
- Parameter for_all_2
+ Parameter for_all_2 :
compat_bool E.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.
(** Specification of [exists] *)
- Parameter exists_1
+ Parameter exists_1 :
compat_bool E.eq f ->
Exists (fun x => f x = true) s -> exists_ f s = true.
- Parameter exists_2
+ Parameter exists_2 :
compat_bool E.eq f ->
exists_ f s = true -> Exists (fun x => f x = true) s.
(** Specification of [partition] *)
- Parameter partition_1 compat_bool E.eq f ->
+ Parameter partition_1 : compat_bool E.eq f ->
fst (partition f s) [=] filter f s.
- Parameter partition_2 compat_bool E.eq f ->
+ Parameter partition_2 : compat_bool E.eq f ->
snd (partition f s) [=] filter (fun x => negb (f x)) s.
(** Specification of [elements] *)
- Parameter elements_1 In x s -> InA E.eq x (elements s).
- Parameter elements_2 InA E.eq x (elements s) -> In x s.
- Parameter elements_3 sort E.lt (elements s).
+ Parameter elements_1 : In x s -> InA E.eq x (elements s).
+ Parameter elements_2 : InA E.eq x (elements s) -> In x s.
+ Parameter elements_3 : sort E.lt (elements s).
(** Specification of [min_elt] *)
- Parameter min_elt_1 min_elt s = Some x -> In x s.
- Parameter min_elt_2 min_elt s = Some x -> In y s -> ~ E.lt y x.
- Parameter min_elt_3 min_elt s = None -> Empty s.
+ Parameter min_elt_1 : min_elt s = Some x -> In x s.
+ Parameter min_elt_2 : min_elt s = Some x -> In y s -> ~ E.lt y x.
+ Parameter min_elt_3 : min_elt s = None -> Empty s.
(** Specification of [max_elt] *)
- Parameter max_elt_1 max_elt s = Some x -> In x s.
- Parameter max_elt_2 max_elt s = Some x -> In y s -> ~ E.lt x y.
- Parameter max_elt_3 max_elt s = None -> Empty s.
+ Parameter max_elt_1 : max_elt s = Some x -> In x s.
+ Parameter max_elt_2 : max_elt s = Some x -> In y s -> ~ E.lt x y.
+ Parameter max_elt_3 : max_elt s = None -> Empty s.
(** Specification of [choose] *)
- Parameter choose_1 choose s = Some x -> In x s.
- Parameter choose_2 choose s = None -> Empty s.
-(* Parameter choose_equal
+ Parameter choose_1 : choose s = Some x -> In x s.
+ Parameter choose_2 : choose s = None -> Empty s.
+(* Parameter choose_equal:
(equal s s')=true -> E.eq (choose s) (choose s'). *)
End Filter.
@@ -293,124 +293,124 @@ End S.
Module Type Sdep.
- Declare Module E OrderedType.
- Definition elt = E.t.
+ Declare Module E : OrderedType.
+ Definition elt := E.t.
- Parameter t Set.
+ Parameter t : Set.
- Parameter In elt -> t -> Prop.
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Add x s s' = forall y, In y s' <-> E.eq x y \/ In y s.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) s := forall x, In x s -> P x.
- Definition Exists (P elt -> Prop) s := exists x, In x s /\ P x.
+ Parameter In : elt -> t -> Prop.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Add x s s' := forall y, In y s' <-> E.eq x y \/ In y s.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
+ Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.
- Notation "s [=] t" = (Equal s t) (at level 70, no associativity).
+ Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
- Definition eq t -> t -> Prop := Equal.
- Parameter lt t -> t -> Prop.
- Parameter compare forall s s' : t, Compare lt eq s s'.
+ Definition eq : t -> t -> Prop := Equal.
+ Parameter lt : t -> t -> Prop.
+ Parameter compare : forall s s' : t, Compare lt eq s s'.
- Parameter eq_refl forall s : t, eq s s.
- Parameter eq_sym forall s s' : t, eq s s' -> eq s' s.
- Parameter eq_trans forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
- Parameter lt_trans forall s s' s'' : t, lt s s' -> lt s' s'' -> lt s s''.
- Parameter lt_not_eq forall s s' : t, lt s s' -> ~ eq s s'.
+ Parameter eq_refl : forall s : t, eq s s.
+ Parameter eq_sym : forall s s' : t, eq s s' -> eq s' s.
+ Parameter eq_trans : forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
+ Parameter lt_trans : forall s s' s'' : t, lt s s' -> lt s' s'' -> lt s s''.
+ Parameter lt_not_eq : forall s s' : t, lt s s' -> ~ eq s s'.
- Parameter eq_In forall (s : t) (x y : elt), E.eq x y -> In x s -> In y s.
+ Parameter eq_In : forall (s : t) (x y : elt), E.eq x y -> In x s -> In y s.
- Parameter empty {s : t | Empty s}.
+ Parameter empty : {s : t | Empty s}.
- Parameter is_empty forall s : t, {Empty s} + {~ Empty s}.
+ Parameter is_empty : forall s : t, {Empty s} + {~ Empty s}.
- Parameter mem forall (x : elt) (s : t), {In x s} + {~ In x s}.
+ Parameter mem : forall (x : elt) (s : t), {In x s} + {~ In x s}.
- Parameter add forall (x : elt) (s : t), {s' : t | Add x s s'}.
+ Parameter add : forall (x : elt) (s : t), {s' : t | Add x s s'}.
Parameter
- singleton forall x : elt, {s : t | forall y : elt, In y s <-> E.eq x y}.
+ singleton : forall x : elt, {s : t | forall y : elt, In y s <-> E.eq x y}.
Parameter
- remove
- forall (x elt) (s : t),
- {s' t | forall y : elt, In y s' <-> ~ E.eq x y /\ In y s}.
+ remove :
+ forall (x : elt) (s : t),
+ {s' : t | forall y : elt, In y s' <-> ~ E.eq x y /\ In y s}.
Parameter
- union
- forall s s' t,
- {s'' t | forall x : elt, In x s'' <-> In x s \/ In x s'}.
+ union :
+ forall s s' : t,
+ {s'' : t | forall x : elt, In x s'' <-> In x s \/ In x s'}.
Parameter
- inter
- forall s s' t,
- {s'' t | forall x : elt, In x s'' <-> In x s /\ In x s'}.
+ inter :
+ forall s s' : t,
+ {s'' : t | forall x : elt, In x s'' <-> In x s /\ In x s'}.
Parameter
- diff
- forall s s' t,
- {s'' t | forall x : elt, In x s'' <-> In x s /\ ~ In x s'}.
+ diff :
+ forall s s' : t,
+ {s'' : t | forall x : elt, In x s'' <-> In x s /\ ~ In x s'}.
- Parameter equal forall s s' : t, {s[=]s'} + {~ s[=]s'}.
+ Parameter equal : forall s s' : t, {s[=]s'} + {~ s[=]s'}.
- Parameter subset forall s s' : t, {Subset s s'} + {~ Subset s s'}.
+ Parameter subset : forall s s' : t, {Subset s s'} + {~ Subset s s'}.
Parameter
- filter
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
- (s t),
- {s' t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.
+ filter :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
+ (s : t),
+ {s' : t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.
Parameter
- for_all
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
- (s t),
+ for_all :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
+ (s : t),
{compat_P E.eq P -> For_all P s} + {compat_P E.eq P -> ~ For_all P s}.
Parameter
- exists_
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
- (s t),
+ exists_ :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
+ (s : t),
{compat_P E.eq P -> Exists P s} + {compat_P E.eq P -> ~ Exists P s}.
Parameter
- partition
- forall (P elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
- (s t),
- {partition t * t |
- let (s1, s2) = partition in
+ partition :
+ forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
+ (s : t),
+ {partition : t * t |
+ let (s1, s2) := partition in
compat_P E.eq P ->
For_all P s1 /\
For_all (fun x => ~ P x) s2 /\
- (forall x elt, In x s <-> In x s1 \/ In x s2)}.
+ (forall x : elt, In x s <-> In x s1 \/ In x s2)}.
Parameter
- elements
- forall s t,
- {l list elt |
- sort E.lt l /\ (forall x elt, In x s <-> InA E.eq x l)}.
+ elements :
+ forall s : t,
+ {l : list elt |
+ sort E.lt l /\ (forall x : elt, In x s <-> InA E.eq x l)}.
Parameter
- fold
- forall (A Set) (f : elt -> A -> A) (s : t) (i : A),
- {r A | let (l,_) := elements s in
+ fold :
+ forall (A : Set) (f : elt -> A -> A) (s : t) (i : A),
+ {r : A | let (l,_) := elements s in
r = fold_left (fun a e => f e a) l i}.
Parameter
- cardinal
- forall s t,
- {r nat | let (l,_) := elements s in r = length l }.
+ cardinal :
+ forall s : t,
+ {r : nat | let (l,_) := elements s in r = length l }.
Parameter
- min_elt
- forall s t,
- {x elt | In x s /\ For_all (fun y => ~ E.lt y x) s} + {Empty s}.
+ min_elt :
+ forall s : t,
+ {x : elt | In x s /\ For_all (fun y => ~ E.lt y x) s} + {Empty s}.
Parameter
- max_elt
- forall s t,
- {x elt | In x s /\ For_all (fun y => ~ E.lt x y) s} + {Empty s}.
+ max_elt :
+ forall s : t,
+ {x : elt | In x s /\ For_all (fun y => ~ E.lt x y) s} + {Empty s}.
- Parameter choose forall s : t, {x : elt | In x s} + {Empty s}.
+ Parameter choose : forall s : t, {x : elt | In x s} + {Empty s}.
End Sdep.
diff --git a/theories/FSets/FSetList.v b/theories/FSets/FSetList.v
index c1b1472ac..444327574 100644
--- a/theories/FSets/FSetList.v
+++ b/theories/FSets/FSetList.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetList.v,v 1.12 2006/03/10 10:49:48 letouzey Exp $ *)
(** * Finite sets library *)
@@ -23,25 +23,25 @@ Unset Strict Implicit.
The specs are proved under the additional condition of being sorted.
And the functions returning sets are proved to preserve this invariant. *)
-Module Raw (X OrderedType).
+Module Raw (X: OrderedType).
- Module E = X.
- Module MX = OrderedTypeFacts X.
+ Module E := X.
+ Module MX := OrderedTypeFacts X.
Import MX.
- Definition elt = X.t.
- Definition t = list elt.
+ Definition elt := X.t.
+ Definition t := list elt.
- Definition empty t := nil.
+ Definition empty : t := nil.
- Definition is_empty (l t) : bool := if l then true else false.
+ Definition is_empty (l : t) : bool := if l then true else false.
(** ** The set operations. *)
- Fixpoint mem (x elt) (s : t) {struct s} : bool :=
+ Fixpoint mem (x : elt) (s : t) {struct s} : bool :=
match s with
| nil => false
- | y : l =>
+ | y :: l =>
match X.compare x y with
| Lt _ => false
| Eq _ => true
@@ -49,83 +49,83 @@ Module Raw (X OrderedType).
end
end.
- Fixpoint add (x elt) (s : t) {struct s} : t :=
+ Fixpoint add (x : elt) (s : t) {struct s} : t :=
match s with
- | nil => x : nil
- | y : l =>
+ | nil => x :: nil
+ | y :: l =>
match X.compare x y with
- | Lt _ => x : s
+ | Lt _ => x :: s
| Eq _ => s
- | Gt _ => y : add x l
+ | Gt _ => y :: add x l
end
end.
- Definition singleton (x elt) : t := x :: nil.
+ Definition singleton (x : elt) : t := x :: nil.
- Fixpoint remove (x elt) (s : t) {struct s} : t :=
+ Fixpoint remove (x : elt) (s : t) {struct s} : t :=
match s with
| nil => nil
- | y : l =>
+ | y :: l =>
match X.compare x y with
| Lt _ => s
| Eq _ => l
- | Gt _ => y : remove x l
+ | Gt _ => y :: remove x l
end
end.
- Fixpoint union (s t) : t -> t :=
+ Fixpoint union (s : t) : t -> t :=
match s with
| nil => fun s' => s'
- | x : l =>
- (fix union_aux (s' t) : t :=
+ | x :: l =>
+ (fix union_aux (s' : t) : t :=
match s' with
| nil => s
- | x' : l' =>
+ | x' :: l' =>
match X.compare x x' with
- | Lt _ => x : union l s'
- | Eq _ => x : union l l'
- | Gt _ => x' : union_aux l'
+ | Lt _ => x :: union l s'
+ | Eq _ => x :: union l l'
+ | Gt _ => x' :: union_aux l'
end
end)
end.
- Fixpoint inter (s t) : t -> t :=
+ Fixpoint inter (s : t) : t -> t :=
match s with
| nil => fun _ => nil
- | x : l =>
- (fix inter_aux (s' t) : t :=
+ | x :: l =>
+ (fix inter_aux (s' : t) : t :=
match s' with
| nil => nil
- | x' : l' =>
+ | x' :: l' =>
match X.compare x x' with
| Lt _ => inter l s'
- | Eq _ => x : inter l l'
+ | Eq _ => x :: inter l l'
| Gt _ => inter_aux l'
end
end)
end.
- Fixpoint diff (s t) : t -> t :=
+ Fixpoint diff (s : t) : t -> t :=
match s with
| nil => fun _ => nil
- | x : l =>
- (fix diff_aux (s' t) : t :=
+ | x :: l =>
+ (fix diff_aux (s' : t) : t :=
match s' with
| nil => s
- | x' : l' =>
+ | x' :: l' =>
match X.compare x x' with
- | Lt _ => x : diff l s'
+ | Lt _ => x :: diff l s'
| Eq _ => diff l l'
| Gt _ => diff_aux l'
end
end)
end.
- Fixpoint equal (s t) : t -> bool :=
- fun s' t =>
+ Fixpoint equal (s : t) : t -> bool :=
+ fun s' : t =>
match s, s' with
| nil, nil => true
- | x : l, x' :: l' =>
+ | x :: l, x' :: l' =>
match X.compare x x' with
| Eq _ => equal l l'
| _ => false
@@ -133,10 +133,10 @@ Module Raw (X OrderedType).
| _, _ => false
end.
- Fixpoint subset (s s' t) {struct s'} : bool :=
+ Fixpoint subset (s s' : t) {struct s'} : bool :=
match s, s' with
| nil, _ => true
- | x : l, x' :: l' =>
+ | x :: l, x' :: l' =>
match X.compare x x' with
| Lt _ => false
| Eq _ => subset l l'
@@ -145,72 +145,72 @@ Module Raw (X OrderedType).
| _, _ => false
end.
- Fixpoint fold (B Set) (f : elt -> B -> B) (s : t) {struct s} :
- B -> B = fun i => match s with
+ Fixpoint fold (B : Set) (f : elt -> B -> B) (s : t) {struct s} :
+ B -> B := fun i => match s with
| nil => i
- | x : l => fold f l (f x i)
+ | x :: l => fold f l (f x i)
end.
- Fixpoint filter (f elt -> bool) (s : t) {struct s} : t :=
+ Fixpoint filter (f : elt -> bool) (s : t) {struct s} : t :=
match s with
| nil => nil
- | x : l => if f x then x :: filter f l else filter f l
+ | x :: l => if f x then x :: filter f l else filter f l
end.
- Fixpoint for_all (f elt -> bool) (s : t) {struct s} : bool :=
+ Fixpoint for_all (f : elt -> bool) (s : t) {struct s} : bool :=
match s with
| nil => true
- | x : l => if f x then for_all f l else false
+ | x :: l => if f x then for_all f l else false
end.
- Fixpoint exists_ (f elt -> bool) (s : t) {struct s} : bool :=
+ Fixpoint exists_ (f : elt -> bool) (s : t) {struct s} : bool :=
match s with
| nil => false
- | x : l => if f x then true else exists_ f l
+ | x :: l => if f x then true else exists_ f l
end.
- Fixpoint partition (f elt -> bool) (s : t) {struct s} :
- t * t =
+ Fixpoint partition (f : elt -> bool) (s : t) {struct s} :
+ t * t :=
match s with
| nil => (nil, nil)
- | x : l =>
- let (s1, s2) = partition f l in
- if f x then (x : s1, s2) else (s1, x :: s2)
+ | x :: l =>
+ let (s1, s2) := partition f l in
+ if f x then (x :: s1, s2) else (s1, x :: s2)
end.
- Definition cardinal (s t) : nat := length s.
+ Definition cardinal (s : t) : nat := length s.
- Definition elements (x t) : list elt := x.
+ Definition elements (x : t) : list elt := x.
- Definition min_elt (s t) : option elt :=
+ Definition min_elt (s : t) : option elt :=
match s with
| nil => None
- | x : _ => Some x
+ | x :: _ => Some x
end.
- Fixpoint max_elt (s t) : option elt :=
+ Fixpoint max_elt (s : t) : option elt :=
match s with
| nil => None
- | x : nil => Some x
- | _ : l => max_elt l
+ | x :: nil => Some x
+ | _ :: l => max_elt l
end.
- Definition choose = min_elt.
+ Definition choose := min_elt.
(** ** Proofs of set operation specifications. *)
- Notation Sort = (sort X.lt).
- Notation Inf = (lelistA X.lt).
- Notation In = (InA X.eq).
+ Notation Sort := (sort X.lt).
+ Notation Inf := (lelistA X.lt).
+ Notation In := (InA X.eq).
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) s := forall x, In x s -> P x.
- Definition Exists (P elt -> Prop) (s : t) := exists x, In x s /\ P x.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
+ Definition Exists (P : elt -> Prop) (s : t) := exists x, In x s /\ P x.
- Lemma mem_1
- forall (s t) (Hs : Sort s) (x : elt), In x s -> mem x s = true.
+ Lemma mem_1 :
+ forall (s : t) (Hs : Sort s) (x : elt), In x s -> mem x s = true.
Proof.
simple induction s; intros.
inversion H.
@@ -221,7 +221,7 @@ Module Raw (X OrderedType).
apply Sort_Inf_In with l; trivial.
Qed.
- Lemma mem_2 forall (s : t) (x : elt), mem x s = true -> In x s.
+ Lemma mem_2 : forall (s : t) (x : elt), mem x s = true -> In x s.
Proof.
simple induction s.
intros; inversion H.
@@ -230,8 +230,8 @@ Module Raw (X OrderedType).
case (X.compare x a); intros; try discriminate; auto.
Qed.
- Lemma add_Inf
- forall (s t) (x a : elt), Inf a s -> X.lt a x -> Inf a (add x s).
+ Lemma add_Inf :
+ forall (s : t) (x a : elt), Inf a s -> X.lt a x -> Inf a (add x s).
Proof.
simple induction s.
simpl; intuition.
@@ -240,7 +240,7 @@ Module Raw (X OrderedType).
Qed.
Hint Resolve add_Inf.
- Lemma add_sort forall (s : t) (Hs : Sort s) (x : elt), Sort (add x s).
+ Lemma add_sort : forall (s : t) (Hs : Sort s) (x : elt), Sort (add x s).
Proof.
simple induction s.
simpl; intuition.
@@ -248,8 +248,8 @@ Module Raw (X OrderedType).
auto.
Qed.
- Lemma add_1
- forall (s t) (Hs : Sort s) (x y : elt), X.eq x y -> In y (add x s).
+ Lemma add_1 :
+ forall (s : t) (Hs : Sort s) (x y : elt), X.eq x y -> In y (add x s).
Proof.
simple induction s.
simpl; intuition.
@@ -257,8 +257,8 @@ Module Raw (X OrderedType).
constructor; apply X.eq_trans with x; auto.
Qed.
- Lemma add_2
- forall (s t) (Hs : Sort s) (x y : elt), In y s -> In y (add x s).
+ Lemma add_2 :
+ forall (s : t) (Hs : Sort s) (x y : elt), In y s -> In y (add x s).
Proof.
simple induction s.
simpl; intuition.
@@ -266,8 +266,8 @@ Module Raw (X OrderedType).
inversion_clear Hs; inversion_clear H0; auto.
Qed.
- Lemma add_3
- forall (s t) (Hs : Sort s) (x y : elt),
+ Lemma add_3 :
+ forall (s : t) (Hs : Sort s) (x y : elt),
~ X.eq x y -> In y (add x s) -> In y s.
Proof.
simple induction s.
@@ -278,8 +278,8 @@ Module Raw (X OrderedType).
constructor 2; apply Hrec with x; auto.
Qed.
- Lemma remove_Inf
- forall (s t) (Hs : Sort s) (x a : elt), Inf a s -> Inf a (remove x s).
+ Lemma remove_Inf :
+ forall (s : t) (Hs : Sort s) (x a : elt), Inf a s -> Inf a (remove x s).
Proof.
simple induction s.
simpl; intuition.
@@ -288,16 +288,16 @@ Module Raw (X OrderedType).
Qed.
Hint Resolve remove_Inf.
- Lemma remove_sort
- forall (s t) (Hs : Sort s) (x : elt), Sort (remove x s).
+ Lemma remove_sort :
+ forall (s : t) (Hs : Sort s) (x : elt), Sort (remove x s).
Proof.
simple induction s.
simpl; intuition.
simpl; intros; case (X.compare x a); intuition; inversion_clear Hs; auto.
Qed.
- Lemma remove_1
- forall (s t) (Hs : Sort s) (x y : elt), X.eq x y -> ~ In y (remove x s).
+ Lemma remove_1 :
+ forall (s : t) (Hs : Sort s) (x y : elt), X.eq x y -> ~ In y (remove x s).
Proof.
simple induction s.
simpl; red; intros; inversion H0.
@@ -311,8 +311,8 @@ Module Raw (X OrderedType).
apply (H H2 _ _ H0 H4).
Qed.
- Lemma remove_2
- forall (s t) (Hs : Sort s) (x y : elt),
+ Lemma remove_2 :
+ forall (s : t) (Hs : Sort s) (x y : elt),
~ X.eq x y -> In y s -> In y (remove x s).
Proof.
simple induction s.
@@ -322,8 +322,8 @@ Module Raw (X OrderedType).
destruct H0; apply X.eq_trans with a; auto.
Qed.
- Lemma remove_3
- forall (s t) (Hs : Sort s) (x y : elt), In y (remove x s) -> In y s.
+ Lemma remove_3 :
+ forall (s : t) (Hs : Sort s) (x y : elt), In y (remove x s) -> In y s.
Proof.
simple induction s.
simpl; intuition.
@@ -332,23 +332,23 @@ Module Raw (X OrderedType).
constructor 2; apply Hrec with x; auto.
Qed.
- Lemma singleton_sort forall x : elt, Sort (singleton x).
+ Lemma singleton_sort : forall x : elt, Sort (singleton x).
Proof.
unfold singleton; simpl; auto.
Qed.
- Lemma singleton_1 forall x y : elt, In y (singleton x) -> X.eq x y.
+ Lemma singleton_1 : forall x y : elt, In y (singleton x) -> X.eq x y.
Proof.
unfold singleton; simpl; intuition.
inversion_clear H; auto; inversion H0.
Qed.
- Lemma singleton_2 forall x y : elt, X.eq x y -> In y (singleton x).
+ Lemma singleton_2 : forall x y : elt, X.eq x y -> In y (singleton x).
Proof.
unfold singleton; simpl; auto.
Qed.
- Ltac DoubleInd =
+ Ltac DoubleInd :=
simple induction s;
[ simpl; auto; try solve [ intros; inversion H ]
| intros x l Hrec; simple induction s';
@@ -356,8 +356,8 @@ Module Raw (X OrderedType).
| intros x' l' Hrec' Hs Hs'; inversion Hs; inversion Hs'; subst;
simpl ] ].
- Lemma union_Inf
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (a : elt),
+ Lemma union_Inf :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (a : elt),
Inf a s -> Inf a s' -> Inf a (union s s').
Proof.
DoubleInd.
@@ -366,34 +366,34 @@ Module Raw (X OrderedType).
Qed.
Hint Resolve union_Inf.
- Lemma union_sort
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'), Sort (union s s').
+ Lemma union_sort :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'), Sort (union s s').
Proof.
DoubleInd; case (X.compare x x'); intuition; constructor; auto.
apply Inf_eq with x'; trivial; apply union_Inf; trivial; apply Inf_eq with x; auto.
- change (Inf x' (union (x : l) l')); auto.
+ change (Inf x' (union (x :: l) l')); auto.
Qed.
- Lemma union_1
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma union_1 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x (union s s') -> In x s \/ In x s'.
Proof.
DoubleInd; case (X.compare x x'); intuition; inversion_clear H; intuition.
- elim (Hrec (x' : l') H1 Hs' x0); intuition.
+ elim (Hrec (x' :: l') H1 Hs' x0); intuition.
elim (Hrec l' H1 H5 x0); intuition.
elim (H0 x0); intuition.
Qed.
- Lemma union_2
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma union_2 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x s -> In x (union s s').
Proof.
DoubleInd.
intros i Hi; case (X.compare x x'); intuition; inversion_clear Hi; auto.
Qed.
- Lemma union_3
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma union_3 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x s' -> In x (union s s').
Proof.
DoubleInd.
@@ -401,8 +401,8 @@ Module Raw (X OrderedType).
constructor; apply X.eq_trans with x'; auto.
Qed.
- Lemma inter_Inf
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (a : elt),
+ Lemma inter_Inf :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (a : elt),
Inf a s -> Inf a s' -> Inf a (inter s s').
Proof.
DoubleInd.
@@ -414,26 +414,26 @@ Module Raw (X OrderedType).
Qed.
Hint Resolve inter_Inf.
- Lemma inter_sort
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'), Sort (inter s s').
+ Lemma inter_sort :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'), Sort (inter s s').
Proof.
DoubleInd; case (X.compare x x'); auto.
constructor; auto.
apply Inf_eq with x'; trivial; apply inter_Inf; trivial; apply Inf_eq with x; auto.
Qed.
- Lemma inter_1
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma inter_1 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x (inter s s') -> In x s.
Proof.
DoubleInd; case (X.compare x x'); intuition.
- constructor 2; apply Hrec with (x':l'); auto.
+ constructor 2; apply Hrec with (x'::l'); auto.
inversion_clear H; auto.
constructor 2; apply Hrec with l'; auto.
Qed.
- Lemma inter_2
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma inter_2 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x (inter s s') -> In x s'.
Proof.
DoubleInd; case (X.compare x x'); intuition; inversion_clear H.
@@ -441,8 +441,8 @@ Module Raw (X OrderedType).
constructor 2; auto.
Qed.
- Lemma inter_3
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma inter_3 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x s -> In x s' -> In x (inter s s').
Proof.
DoubleInd.
@@ -454,13 +454,13 @@ Module Raw (X OrderedType).
inversion_clear His; auto; inversion_clear His'; auto.
constructor; apply X.eq_trans with x'; auto.
- change (In i (inter (x : l) l')).
+ change (In i (inter (x :: l) l')).
inversion_clear His'; auto.
generalize (Sort_Inf_In Hs (cons_leA _ _ _ _ l0) His); order.
Qed.
- Lemma diff_Inf
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (a : elt),
+ Lemma diff_Inf :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (a : elt),
Inf a s -> Inf a s' -> Inf a (diff s s').
Proof.
DoubleInd.
@@ -474,24 +474,24 @@ Module Raw (X OrderedType).
Qed.
Hint Resolve diff_Inf.
- Lemma diff_sort
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'), Sort (diff s s').
+ Lemma diff_sort :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'), Sort (diff s s').
Proof.
DoubleInd; case (X.compare x x'); auto.
Qed.
- Lemma diff_1
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma diff_1 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x (diff s s') -> In x s.
Proof.
DoubleInd; case (X.compare x x'); intuition.
inversion_clear H; auto.
- constructor 2; apply Hrec with (x':l'); auto.
+ constructor 2; apply Hrec with (x'::l'); auto.
constructor 2; apply Hrec with l'; auto.
Qed.
- Lemma diff_2
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma diff_2 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x (diff s s') -> ~ In x s'.
Proof.
DoubleInd.
@@ -500,7 +500,7 @@ Module Raw (X OrderedType).
inversion_clear H.
generalize (Sort_Inf_In Hs' (cons_leA _ _ _ _ l0) H3); order.
- apply Hrec with (x':l') x0; auto.
+ apply Hrec with (x'::l') x0; auto.
inversion_clear H3.
generalize (Sort_Inf_In H1 H2 (diff_1 H1 H5 H)); order.
@@ -511,8 +511,8 @@ Module Raw (X OrderedType).
apply H0 with x0; auto.
Qed.
- Lemma diff_3
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
+ Lemma diff_3 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s') (x : elt),
In x s -> ~ In x s' -> In x (diff s s').
Proof.
DoubleInd.
@@ -520,38 +520,38 @@ Module Raw (X OrderedType).
elim His'; constructor; apply X.eq_trans with x; auto.
Qed.
- Lemma equal_1
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'),
+ Lemma equal_1 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'),
Equal s s' -> equal s s' = true.
Proof.
simple induction s; unfold Equal.
intro s'; case s'; auto.
simpl; intuition.
- elim (H e); intros; assert (A In e nil); auto; inversion A.
+ elim (H e); intros; assert (A : In e nil); auto; inversion A.
intros x l Hrec s'.
case s'.
- intros; elim (H x); intros; assert (A In x nil); auto; inversion A.
+ intros; elim (H x); intros; assert (A : In x nil); auto; inversion A.
intros x' l' Hs Hs'; inversion Hs; inversion Hs'; subst.
simpl; case (X.compare x); intros; auto.
elim (H x); intros.
- assert (A In x (x' :: l')); auto; inversion_clear A.
+ assert (A : In x (x' :: l')); auto; inversion_clear A.
order.
generalize (Sort_Inf_In H5 H6 H4); order.
apply Hrec; intuition; elim (H a); intros.
- assert (A In a (x' :: l')); auto; inversion_clear A; auto.
+ assert (A : In a (x' :: l')); auto; inversion_clear A; auto.
generalize (Sort_Inf_In H1 H2 H0); order.
- assert (A In a (x :: l)); auto; inversion_clear A; auto.
+ assert (A : In a (x :: l)); auto; inversion_clear A; auto.
generalize (Sort_Inf_In H5 H6 H0); order.
elim (H x'); intros.
- assert (A In x' (x :: l)); auto; inversion_clear A.
+ assert (A : In x' (x :: l)); auto; inversion_clear A.
order.
generalize (Sort_Inf_In H1 H2 H4); order.
Qed.
- Lemma equal_2 forall s s' : t, equal s s' = true -> Equal s s'.
+ Lemma equal_2 : forall s s' : t, equal s s' = true -> Equal s s'.
Proof.
simple induction s; unfold Equal.
intro s'; case s'; intros.
@@ -566,35 +566,35 @@ Module Raw (X OrderedType).
constructor; apply X.eq_trans with x'; auto.
Qed.
- Lemma subset_1
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'),
+ Lemma subset_1 :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'),
Subset s s' -> subset s s' = true.
Proof.
intros s s'; generalize s' s; clear s s'.
simple induction s'; unfold Subset.
intro s; case s; auto.
- intros; elim (H e); intros; assert (A In e nil); auto; inversion A.
+ intros; elim (H e); intros; assert (A : In e nil); auto; inversion A.
intros x' l' Hrec s; case s.
simpl; auto.
intros x l Hs Hs'; inversion Hs; inversion Hs'; subst.
simpl; case (X.compare x); intros; auto.
- assert (A In x (x' :: l')); auto; inversion_clear A.
+ assert (A : In x (x' :: l')); auto; inversion_clear A.
order.
generalize (Sort_Inf_In H5 H6 H0); order.
apply Hrec; intuition.
- assert (A In a (x' :: l')); auto; inversion_clear A; auto.
+ assert (A : In a (x' :: l')); auto; inversion_clear A; auto.
generalize (Sort_Inf_In H1 H2 H0); order.
apply Hrec; intuition.
- assert (A In a (x' :: l')); auto; inversion_clear A; auto.
+ assert (A : In a (x' :: l')); auto; inversion_clear A; auto.
inversion_clear H0.
order.
generalize (Sort_Inf_In H1 H2 H4); order.
Qed.
- Lemma subset_2 forall s s' : t, subset s s' = true -> Subset s s'.
+ Lemma subset_2 : forall s s' : t, subset s s' = true -> Subset s s'.
Proof.
intros s s'; generalize s' s; clear s s'.
simple induction s'; unfold Subset.
@@ -607,53 +607,53 @@ Module Raw (X OrderedType).
inversion_clear H0.
constructor; apply X.eq_trans with x; auto.
constructor 2; apply Hrec with l; auto.
- constructor 2; apply Hrec with (x:l); auto.
+ constructor 2; apply Hrec with (x::l); auto.
Qed.
- Lemma empty_sort Sort empty.
+ Lemma empty_sort : Sort empty.
Proof.
unfold empty; constructor.
Qed.
- Lemma empty_1 Empty empty.
+ Lemma empty_1 : Empty empty.
Proof.
unfold Empty, empty; intuition; inversion H.
Qed.
- Lemma is_empty_1 forall s : t, Empty s -> is_empty s = true.
+ Lemma is_empty_1 : forall s : t, Empty s -> is_empty s = true.
Proof.
unfold Empty; intro s; case s; simpl; intuition.
elim (H e); auto.
Qed.
- Lemma is_empty_2 forall s : t, is_empty s = true -> Empty s.
+ Lemma is_empty_2 : forall s : t, is_empty s = true -> Empty s.
Proof.
unfold Empty; intro s; case s; simpl; intuition;
inversion H0.
Qed.
- Lemma elements_1 forall (s : t) (x : elt), In x s -> In x (elements s).
+ Lemma elements_1 : forall (s : t) (x : elt), In x s -> In x (elements s).
Proof.
unfold elements; auto.
Qed.
- Lemma elements_2 forall (s : t) (x : elt), In x (elements s) -> In x s.
+ Lemma elements_2 : forall (s : t) (x : elt), In x (elements s) -> In x s.
Proof.
unfold elements; auto.
Qed.
- Lemma elements_3 forall (s : t) (Hs : Sort s), Sort (elements s).
+ Lemma elements_3 : forall (s : t) (Hs : Sort s), Sort (elements s).
Proof.
unfold elements; auto.
Qed.
- Lemma min_elt_1 forall (s : t) (x : elt), min_elt s = Some x -> In x s.
+ Lemma min_elt_1 : forall (s : t) (x : elt), min_elt s = Some x -> In x s.
Proof.
intro s; case s; simpl; intros; inversion H; auto.
Qed.
- Lemma min_elt_2
- forall (s t) (Hs : Sort s) (x y : elt),
+ Lemma min_elt_2 :
+ forall (s : t) (Hs : Sort s) (x y : elt),
min_elt s = Some x -> In y s -> ~ X.lt y x.
Proof.
simple induction s; simpl.
@@ -667,13 +667,13 @@ Module Raw (X OrderedType).
generalize (H H1 e y (refl_equal (Some e)) H2); order.
Qed.
- Lemma min_elt_3 forall s : t, min_elt s = None -> Empty s.
+ Lemma min_elt_3 : forall s : t, min_elt s = None -> Empty s.
Proof.
unfold Empty; intro s; case s; simpl; intuition;
inversion H; inversion H0.
Qed.
- Lemma max_elt_1 forall (s : t) (x : elt), max_elt s = Some x -> In x s.
+ Lemma max_elt_1 : forall (s : t) (x : elt), max_elt s = Some x -> In x s.
Proof.
simple induction s; simpl.
intros; inversion H.
@@ -684,8 +684,8 @@ Module Raw (X OrderedType).
constructor 2; apply (H _ H0).
Qed.
- Lemma max_elt_2
- forall (s t) (Hs : Sort s) (x y : elt),
+ Lemma max_elt_2 :
+ forall (s : t) (Hs : Sort s) (x y : elt),
max_elt s = Some x -> In y s -> ~ X.lt x y.
Proof.
simple induction s; simpl.
@@ -697,12 +697,12 @@ Module Raw (X OrderedType).
order.
inversion H3.
intros; inversion_clear Hs; inversion_clear H3; inversion_clear H1.
- assert (In e (e:l0)) by auto.
+ assert (In e (e::l0)) by auto.
generalize (H H2 x0 e H0 H1); order.
generalize (H H2 x0 y H0 H3); order.
Qed.
- Lemma max_elt_3 forall s : t, max_elt s = None -> Empty s.
+ Lemma max_elt_3 : forall s : t, max_elt s = None -> Empty s.
Proof.
unfold Empty; simple induction s; simpl.
red; intros; inversion H0.
@@ -711,13 +711,13 @@ Module Raw (X OrderedType).
elim (H H0 e); auto.
Qed.
- Definition choose_1
- forall (s t) (x : elt), choose s = Some x -> In x s := min_elt_1.
+ Definition choose_1 :
+ forall (s : t) (x : elt), choose s = Some x -> In x s := min_elt_1.
- Definition choose_2 forall s : t, choose s = None -> Empty s := min_elt_3.
+ Definition choose_2 : forall s : t, choose s = None -> Empty s := min_elt_3.
- Lemma fold_1
- forall (s t) (Hs : Sort s) (A : Set) (i : A) (f : elt -> A -> A),
+ Lemma fold_1 :
+ forall (s : t) (Hs : Sort s) (A : Set) (i : A) (f : elt -> A -> A),
fold f s i = fold_left (fun a e => f e a) (elements s) i.
Proof.
induction s.
@@ -727,15 +727,15 @@ Module Raw (X OrderedType).
simpl; auto.
Qed.
- Lemma cardinal_1
- forall (s t) (Hs : Sort s),
+ Lemma cardinal_1 :
+ forall (s : t) (Hs : Sort s),
cardinal s = length (elements s).
Proof.
auto.
Qed.
- Lemma filter_Inf
- forall (s t) (Hs : Sort s) (x : elt) (f : elt -> bool),
+ Lemma filter_Inf :
+ forall (s : t) (Hs : Sort s) (x : elt) (f : elt -> bool),
Inf x s -> Inf x (filter f s).
Proof.
simple induction s; simpl.
@@ -747,8 +747,8 @@ Module Raw (X OrderedType).
apply Inf_lt with x; auto.
Qed.
- Lemma filter_sort
- forall (s t) (Hs : Sort s) (f : elt -> bool), Sort (filter f s).
+ Lemma filter_sort :
+ forall (s : t) (Hs : Sort s) (f : elt -> bool), Sort (filter f s).
Proof.
simple induction s; simpl.
auto.
@@ -758,8 +758,8 @@ Module Raw (X OrderedType).
apply filter_Inf; auto.
Qed.
- Lemma filter_1
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_1 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool X.eq f -> In x (filter f s) -> In x s.
Proof.
simple induction s; simpl.
@@ -772,8 +772,8 @@ Module Raw (X OrderedType).
constructor 2; apply (Hrec a f Hf); trivial.
Qed.
- Lemma filter_2
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_2 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool X.eq f -> In x (filter f s) -> f x = true.
Proof.
simple induction s; simpl.
@@ -784,8 +784,8 @@ Module Raw (X OrderedType).
symmetry; auto.
Qed.
- Lemma filter_3
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_3 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool X.eq f -> In x s -> f x = true -> In x (filter f s).
Proof.
simple induction s; simpl.
@@ -797,8 +797,8 @@ Module Raw (X OrderedType).
rewrite <- (H a (X.eq_sym H1)); intros; discriminate.
Qed.
- Lemma for_all_1
- forall (s t) (f : elt -> bool),
+ Lemma for_all_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
Proof.
@@ -809,8 +809,8 @@ Module Raw (X OrderedType).
intros; rewrite (H x); auto.
Qed.
- Lemma for_all_2
- forall (s t) (f : elt -> bool),
+ Lemma for_all_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.
Proof.
@@ -825,8 +825,8 @@ Module Raw (X OrderedType).
rewrite (Hf a x); auto.
Qed.
- Lemma exists_1
- forall (s t) (f : elt -> bool),
+ Lemma exists_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f -> Exists (fun x => f x = true) s -> exists_ f s = true.
Proof.
simple induction s; simpl; auto; unfold Exists.
@@ -843,8 +843,8 @@ Module Raw (X OrderedType).
exists a; auto.
Qed.
- Lemma exists_2
- forall (s t) (f : elt -> bool),
+ Lemma exists_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f -> exists_ f s = true -> Exists (fun x => f x = true) s.
Proof.
simple induction s; simpl; auto; unfold Exists.
@@ -856,8 +856,8 @@ Module Raw (X OrderedType).
exists a; auto.
Qed.
- Lemma partition_Inf_1
- forall (s t) (Hs : Sort s) (f : elt -> bool) (x : elt),
+ Lemma partition_Inf_1 :
+ forall (s : t) (Hs : Sort s) (f : elt -> bool) (x : elt),
Inf x s -> Inf x (fst (partition f s)).
Proof.
simple induction s; simpl.
@@ -869,8 +869,8 @@ Module Raw (X OrderedType).
intros; apply H2; apply Inf_lt with x; auto.
Qed.
- Lemma partition_Inf_2
- forall (s t) (Hs : Sort s) (f : elt -> bool) (x : elt),
+ Lemma partition_Inf_2 :
+ forall (s : t) (Hs : Sort s) (f : elt -> bool) (x : elt),
Inf x s -> Inf x (snd (partition f s)).
Proof.
simple induction s; simpl.
@@ -882,8 +882,8 @@ Module Raw (X OrderedType).
auto.
Qed.
- Lemma partition_sort_1
- forall (s t) (Hs : Sort s) (f : elt -> bool), Sort (fst (partition f s)).
+ Lemma partition_sort_1 :
+ forall (s : t) (Hs : Sort s) (f : elt -> bool), Sort (fst (partition f s)).
Proof.
simple induction s; simpl.
auto.
@@ -892,8 +892,8 @@ Module Raw (X OrderedType).
case (f x); case (partition f l); simpl; auto.
Qed.
- Lemma partition_sort_2
- forall (s t) (Hs : Sort s) (f : elt -> bool), Sort (snd (partition f s)).
+ Lemma partition_sort_2 :
+ forall (s : t) (Hs : Sort s) (f : elt -> bool), Sort (snd (partition f s)).
Proof.
simple induction s; simpl.
auto.
@@ -902,8 +902,8 @@ Module Raw (X OrderedType).
case (f x); case (partition f l); simpl; auto.
Qed.
- Lemma partition_1
- forall (s t) (f : elt -> bool),
+ Lemma partition_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f -> Equal (fst (partition f s)) (filter f s).
Proof.
simple induction s; simpl; auto; unfold Equal.
@@ -917,8 +917,8 @@ Module Raw (X OrderedType).
constructor 2; rewrite H; auto.
Qed.
- Lemma partition_2
- forall (s t) (f : elt -> bool),
+ Lemma partition_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f ->
Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
Proof.
@@ -933,33 +933,33 @@ Module Raw (X OrderedType).
constructor 2; rewrite H; auto.
Qed.
- Definition eq t -> t -> Prop := Equal.
+ Definition eq : t -> t -> Prop := Equal.
- Lemma eq_refl forall s : t, eq s s.
+ Lemma eq_refl : forall s : t, eq s s.
Proof.
unfold eq, Equal; intuition.
Qed.
- Lemma eq_sym forall s s' : t, eq s s' -> eq s' s.
+ Lemma eq_sym : forall s s' : t, eq s s' -> eq s' s.
Proof.
unfold eq, Equal; intros; destruct (H a); intuition.
Qed.
- Lemma eq_trans forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
+ Lemma eq_trans : forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
Proof.
unfold eq, Equal; intros; destruct (H a); destruct (H0 a); intuition.
Qed.
- Inductive lt t -> t -> Prop :=
- | lt_nil forall (x : elt) (s : t), lt nil (x :: s)
- | lt_cons_lt
- forall (x y elt) (s s' : t), X.lt x y -> lt (x :: s) (y :: s')
- | lt_cons_eq
- forall (x y elt) (s s' : t),
- X.eq x y -> lt s s' -> lt (x : s) (y :: s').
+ Inductive lt : t -> t -> Prop :=
+ | lt_nil : forall (x : elt) (s : t), lt nil (x :: s)
+ | lt_cons_lt :
+ forall (x y : elt) (s s' : t), X.lt x y -> lt (x :: s) (y :: s')
+ | lt_cons_eq :
+ forall (x y : elt) (s s' : t),
+ X.eq x y -> lt s s' -> lt (x :: s) (y :: s').
Hint Constructors lt.
- Lemma lt_trans forall s s' s'' : t, lt s s' -> lt s' s'' -> lt s s''.
+ Lemma lt_trans : forall s s' s'' : t, lt s s' -> lt s' s'' -> lt s s''.
Proof.
intros s s' s'' H; generalize s''; clear s''; elim H.
intros x l s'' H'; inversion_clear H'; auto.
@@ -972,32 +972,32 @@ Module Raw (X OrderedType).
constructor 3; auto; apply X.eq_trans with y; auto.
Qed.
- Lemma lt_not_eq
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'), lt s s' -> ~ eq s s'.
+ Lemma lt_not_eq :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'), lt s s' -> ~ eq s s'.
Proof.
unfold eq, Equal.
intros s s' Hs Hs' H; generalize Hs Hs'; clear Hs Hs'; elim H; intros; intro.
elim (H0 x); intros.
- assert (X In x nil); auto; inversion X.
+ assert (X : In x nil); auto; inversion X.
inversion_clear Hs; inversion_clear Hs'.
elim (H1 x); intros.
- assert (X In x (y :: s'0)); auto; inversion_clear X.
+ assert (X : In x (y :: s'0)); auto; inversion_clear X.
order.
generalize (Sort_Inf_In H4 H5 H8); order.
inversion_clear Hs; inversion_clear Hs'.
elim H2; auto; split; intros.
generalize (Sort_Inf_In H4 H5 H8); intros.
elim (H3 a); intros.
- assert (X In a (y :: s'0)); auto; inversion_clear X; auto.
+ assert (X : In a (y :: s'0)); auto; inversion_clear X; auto.
order.
generalize (Sort_Inf_In H6 H7 H8); intros.
elim (H3 a); intros.
- assert (X In a (x :: s0)); auto; inversion_clear X; auto.
+ assert (X : In a (x :: s0)); auto; inversion_clear X; auto.
order.
Qed.
- Definition compare
- forall (s s' t) (Hs : Sort s) (Hs' : Sort s'), Compare lt eq s s'.
+ Definition compare :
+ forall (s s' : t) (Hs : Sort s) (Hs' : Sort s'), Compare lt eq s s'.
Proof.
simple induction s.
intros; case s'.
@@ -1027,133 +1027,133 @@ End Raw.
Now, in order to really provide a functor implementing [S], we
need to encapsulate everything into a type of strictly ordered lists. *)
-Module Make (X OrderedType) <: S with Module E := X.
+Module Make (X: OrderedType) <: S with Module E := X.
- Module E = X.
- Module Raw = Raw X.
+ Module E := X.
+ Module Raw := Raw X.
- Record slist Set := {this :> Raw.t; sorted : sort X.lt this}.
- Definition t = slist.
- Definition elt = X.t.
+ Record slist : Set := {this :> Raw.t; sorted : sort X.lt this}.
+ Definition t := slist.
+ Definition elt := X.t.
- Definition In (x elt) (s : t) := InA X.eq x s.(this).
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) s := forall x, In x s -> P x.
- Definition Exists (P elt -> Prop) s := exists x, In x s /\ P x.
+ Definition In (x : elt) (s : t) := InA X.eq x s.(this).
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
+ Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.
- Definition In_1 (s t) := Raw.MX.In_eq (l:=s.(this)).
+ Definition In_1 (s : t) := Raw.MX.In_eq (l:=s.(this)).
- Definition mem (x elt) (s : t) := Raw.mem x s.
- Definition mem_1 (s t) := Raw.mem_1 (sorted s).
- Definition mem_2 (s t) := Raw.mem_2 (s:=s).
-
- Definition add x s = Build_slist (Raw.add_sort (sorted s) x).
- Definition add_1 (s t) := Raw.add_1 (sorted s).
- Definition add_2 (s t) := Raw.add_2 (sorted s).
- Definition add_3 (s t) := Raw.add_3 (sorted s).
-
- Definition remove x s = Build_slist (Raw.remove_sort (sorted s) x).
- Definition remove_1 (s t) := Raw.remove_1 (sorted s).
- Definition remove_2 (s t) := Raw.remove_2 (sorted s).
- Definition remove_3 (s t) := Raw.remove_3 (sorted s).
+ Definition mem (x : elt) (s : t) := Raw.mem x s.
+ Definition mem_1 (s : t) := Raw.mem_1 (sorted s).
+ Definition mem_2 (s : t) := Raw.mem_2 (s:=s).
+
+ Definition add x s := Build_slist (Raw.add_sort (sorted s) x).
+ Definition add_1 (s : t) := Raw.add_1 (sorted s).
+ Definition add_2 (s : t) := Raw.add_2 (sorted s).
+ Definition add_3 (s : t) := Raw.add_3 (sorted s).
+
+ Definition remove x s := Build_slist (Raw.remove_sort (sorted s) x).
+ Definition remove_1 (s : t) := Raw.remove_1 (sorted s).
+ Definition remove_2 (s : t) := Raw.remove_2 (sorted s).
+ Definition remove_3 (s : t) := Raw.remove_3 (sorted s).
- Definition singleton x = Build_slist (Raw.singleton_sort x).
- Definition singleton_1 = Raw.singleton_1.
- Definition singleton_2 = Raw.singleton_2.
+ Definition singleton x := Build_slist (Raw.singleton_sort x).
+ Definition singleton_1 := Raw.singleton_1.
+ Definition singleton_2 := Raw.singleton_2.
- Definition union (s s' t) :=
+ Definition union (s s' : t) :=
Build_slist (Raw.union_sort (sorted s) (sorted s')).
- Definition union_1 (s s' t) := Raw.union_1 (sorted s) (sorted s').
- Definition union_2 (s s' t) := Raw.union_2 (sorted s) (sorted s').
- Definition union_3 (s s' t) := Raw.union_3 (sorted s) (sorted s').
+ Definition union_1 (s s' : t) := Raw.union_1 (sorted s) (sorted s').
+ Definition union_2 (s s' : t) := Raw.union_2 (sorted s) (sorted s').
+ Definition union_3 (s s' : t) := Raw.union_3 (sorted s) (sorted s').
- Definition inter (s s' t) :=
+ Definition inter (s s' : t) :=
Build_slist (Raw.inter_sort (sorted s) (sorted s')).
- Definition inter_1 (s s' t) := Raw.inter_1 (sorted s) (sorted s').
- Definition inter_2 (s s' t) := Raw.inter_2 (sorted s) (sorted s').
- Definition inter_3 (s s' t) := Raw.inter_3 (sorted s) (sorted s').
+ Definition inter_1 (s s' : t) := Raw.inter_1 (sorted s) (sorted s').
+ Definition inter_2 (s s' : t) := Raw.inter_2 (sorted s) (sorted s').
+ Definition inter_3 (s s' : t) := Raw.inter_3 (sorted s) (sorted s').
- Definition diff (s s' t) :=
+ Definition diff (s s' : t) :=
Build_slist (Raw.diff_sort (sorted s) (sorted s')).
- Definition diff_1 (s s' t) := Raw.diff_1 (sorted s) (sorted s').
- Definition diff_2 (s s' t) := Raw.diff_2 (sorted s) (sorted s').
- Definition diff_3 (s s' t) := Raw.diff_3 (sorted s) (sorted s').
+ Definition diff_1 (s s' : t) := Raw.diff_1 (sorted s) (sorted s').
+ Definition diff_2 (s s' : t) := Raw.diff_2 (sorted s) (sorted s').
+ Definition diff_3 (s s' : t) := Raw.diff_3 (sorted s) (sorted s').
- Definition equal (s s' t) := Raw.equal s s'.
- Definition equal_1 (s s' t) := Raw.equal_1 (sorted s) (sorted s').
- Definition equal_2 (s s' t) := Raw.equal_2 (s:=s) (s':=s').
+ Definition equal (s s' : t) := Raw.equal s s'.
+ Definition equal_1 (s s' : t) := Raw.equal_1 (sorted s) (sorted s').
+ Definition equal_2 (s s' : t) := Raw.equal_2 (s:=s) (s':=s').
- Definition subset (s s' t) := Raw.subset s s'.
- Definition subset_1 (s s' t) := Raw.subset_1 (sorted s) (sorted s').
- Definition subset_2 (s s' t) := Raw.subset_2 (s:=s) (s':=s').
+ Definition subset (s s' : t) := Raw.subset s s'.
+ Definition subset_1 (s s' : t) := Raw.subset_1 (sorted s) (sorted s').
+ Definition subset_2 (s s' : t) := Raw.subset_2 (s:=s) (s':=s').
- Definition empty = Build_slist Raw.empty_sort.
- Definition empty_1 = Raw.empty_1.
+ Definition empty := Build_slist Raw.empty_sort.
+ Definition empty_1 := Raw.empty_1.
- Definition is_empty (s t) := Raw.is_empty s.
- Definition is_empty_1 (s t) := Raw.is_empty_1 (s:=s).
- Definition is_empty_2 (s t) := Raw.is_empty_2 (s:=s).
-
- Definition elements (s t) := Raw.elements s.
- Definition elements_1 (s t) := Raw.elements_1 (s:=s).
- Definition elements_2 (s t) := Raw.elements_2 (s:=s).
- Definition elements_3 (s t) := Raw.elements_3 (sorted s).
+ Definition is_empty (s : t) := Raw.is_empty s.
+ Definition is_empty_1 (s : t) := Raw.is_empty_1 (s:=s).
+ Definition is_empty_2 (s : t) := Raw.is_empty_2 (s:=s).
+
+ Definition elements (s : t) := Raw.elements s.
+ Definition elements_1 (s : t) := Raw.elements_1 (s:=s).
+ Definition elements_2 (s : t) := Raw.elements_2 (s:=s).
+ Definition elements_3 (s : t) := Raw.elements_3 (sorted s).
- Definition min_elt (s t) := Raw.min_elt s.
- Definition min_elt_1 (s t) := Raw.min_elt_1 (s:=s).
- Definition min_elt_2 (s t) := Raw.min_elt_2 (sorted s).
- Definition min_elt_3 (s t) := Raw.min_elt_3 (s:=s).
-
- Definition max_elt (s t) := Raw.max_elt s.
- Definition max_elt_1 (s t) := Raw.max_elt_1 (s:=s).
- Definition max_elt_2 (s t) := Raw.max_elt_2 (sorted s).
- Definition max_elt_3 (s t) := Raw.max_elt_3 (s:=s).
+ Definition min_elt (s : t) := Raw.min_elt s.
+ Definition min_elt_1 (s : t) := Raw.min_elt_1 (s:=s).
+ Definition min_elt_2 (s : t) := Raw.min_elt_2 (sorted s).
+ Definition min_elt_3 (s : t) := Raw.min_elt_3 (s:=s).
+
+ Definition max_elt (s : t) := Raw.max_elt s.
+ Definition max_elt_1 (s : t) := Raw.max_elt_1 (s:=s).
+ Definition max_elt_2 (s : t) := Raw.max_elt_2 (sorted s).
+ Definition max_elt_3 (s : t) := Raw.max_elt_3 (s:=s).
- Definition choose = min_elt.
- Definition choose_1 = min_elt_1.
- Definition choose_2 = min_elt_3.
+ Definition choose := min_elt.
+ Definition choose_1 := min_elt_1.
+ Definition choose_2 := min_elt_3.
- Definition fold (B Set) (f : elt -> B -> B) (s : t) := Raw.fold (B:=B) f s.
- Definition fold_1 (s t) := Raw.fold_1 (sorted s).
+ Definition fold (B : Set) (f : elt -> B -> B) (s : t) := Raw.fold (B:=B) f s.
+ Definition fold_1 (s : t) := Raw.fold_1 (sorted s).
- Definition cardinal (s t) := Raw.cardinal s.
- Definition cardinal_1 (s t) := Raw.cardinal_1 (sorted s).
+ Definition cardinal (s : t) := Raw.cardinal s.
+ Definition cardinal_1 (s : t) := Raw.cardinal_1 (sorted s).
- Definition filter (f elt -> bool) (s : t) :=
+ Definition filter (f : elt -> bool) (s : t) :=
Build_slist (Raw.filter_sort (sorted s) f).
- Definition filter_1 (s t) := Raw.filter_1 (s:=s).
- Definition filter_2 (s t) := Raw.filter_2 (s:=s).
- Definition filter_3 (s t) := Raw.filter_3 (s:=s).
+ Definition filter_1 (s : t) := Raw.filter_1 (s:=s).
+ Definition filter_2 (s : t) := Raw.filter_2 (s:=s).
+ Definition filter_3 (s : t) := Raw.filter_3 (s:=s).
- Definition for_all (f elt -> bool) (s : t) := Raw.for_all f s.
- Definition for_all_1 (s t) := Raw.for_all_1 (s:=s).
- Definition for_all_2 (s t) := Raw.for_all_2 (s:=s).
-
- Definition exists_ (f elt -> bool) (s : t) := Raw.exists_ f s.
- Definition exists_1 (s t) := Raw.exists_1 (s:=s).
- Definition exists_2 (s t) := Raw.exists_2 (s:=s).
-
- Definition partition (f elt -> bool) (s : t) :=
- let p = Raw.partition f s in
- (Build_slist (this=fst p) (Raw.partition_sort_1 (sorted s) f),
- Build_slist (this=snd p) (Raw.partition_sort_2 (sorted s) f)).
- Definition partition_1 (s t) := Raw.partition_1 s.
- Definition partition_2 (s t) := Raw.partition_2 s.
-
- Definition eq (s s' t) := Raw.eq s s'.
- Definition eq_refl (s t) := Raw.eq_refl s.
- Definition eq_sym (s s' t) := Raw.eq_sym (s:=s) (s':=s').
- Definition eq_trans (s s' s'' t) :=
- Raw.eq_trans (s=s) (s':=s') (s'':=s'').
+ Definition for_all (f : elt -> bool) (s : t) := Raw.for_all f s.
+ Definition for_all_1 (s : t) := Raw.for_all_1 (s:=s).
+ Definition for_all_2 (s : t) := Raw.for_all_2 (s:=s).
+
+ Definition exists_ (f : elt -> bool) (s : t) := Raw.exists_ f s.
+ Definition exists_1 (s : t) := Raw.exists_1 (s:=s).
+ Definition exists_2 (s : t) := Raw.exists_2 (s:=s).
+
+ Definition partition (f : elt -> bool) (s : t) :=
+ let p := Raw.partition f s in
+ (Build_slist (this:=fst p) (Raw.partition_sort_1 (sorted s) f),
+ Build_slist (this:=snd p) (Raw.partition_sort_2 (sorted s) f)).
+ Definition partition_1 (s : t) := Raw.partition_1 s.
+ Definition partition_2 (s : t) := Raw.partition_2 s.
+
+ Definition eq (s s' : t) := Raw.eq s s'.
+ Definition eq_refl (s : t) := Raw.eq_refl s.
+ Definition eq_sym (s s' : t) := Raw.eq_sym (s:=s) (s':=s').
+ Definition eq_trans (s s' s'' : t) :=
+ Raw.eq_trans (s:=s) (s':=s') (s'':=s'').
- Definition lt (s s' t) := Raw.lt s s'.
- Definition lt_trans (s s' s'' t) :=
- Raw.lt_trans (s=s) (s':=s') (s'':=s'').
- Definition lt_not_eq (s s' t) := Raw.lt_not_eq (sorted s) (sorted s').
+ Definition lt (s s' : t) := Raw.lt s s'.
+ Definition lt_trans (s s' s'' : t) :=
+ Raw.lt_trans (s:=s) (s':=s') (s'':=s'').
+ Definition lt_not_eq (s s' : t) := Raw.lt_not_eq (sorted s) (sorted s').
- Definition compare forall s s' : t, Compare lt eq s s'.
+ Definition compare : forall s s' : t, Compare lt eq s s'.
Proof.
intros; elim (Raw.compare (sorted s) (sorted s'));
[ constructor 1 | constructor 2 | constructor 3 ];
diff --git a/theories/FSets/FSetProperties.v b/theories/FSets/FSetProperties.v
index afdc20250..b67c1245e 100644
--- a/theories/FSets/FSetProperties.v
+++ b/theories/FSets/FSetProperties.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetProperties.v,v 1.16 2006/03/13 04:59:24 letouzey Exp $ *)
(** * Finite sets library *)
@@ -22,47 +22,47 @@ Set Implicit Arguments.
Unset Strict Implicit.
Section Misc.
-Variable A B Set.
-Variable eqA A -> A -> Prop.
-Variable eqB B -> B -> Prop.
+Variable A B : Set.
+Variable eqA : A -> A -> Prop.
+Variable eqB : B -> B -> Prop.
(** Two-argument functions that allow to reorder its arguments. *)
-Definition transpose (f A -> B -> B) :=
- forall (x y A) (z : B), eqB (f x (f y z)) (f y (f x z)).
+Definition transpose (f : A -> B -> B) :=
+ forall (x y : A) (z : B), eqB (f x (f y z)) (f y (f x z)).
(** Compatibility of a two-argument function with respect to two equalities. *)
-Definition compat_op (f A -> B -> B) :=
- forall (x x' A) (y y' : B), eqA x x' -> eqB y y' -> eqB (f x y) (f x' y').
+Definition compat_op (f : A -> B -> B) :=
+ forall (x x' : A) (y y' : B), eqA x x' -> eqB y y' -> eqB (f x y) (f x' y').
(** Compatibility of a function upon natural numbers. *)
-Definition compat_nat (f A -> nat) :=
- forall x x' A, eqA x x' -> f x = f x'.
+Definition compat_nat (f : A -> nat) :=
+ forall x x' : A, eqA x x' -> f x = f x'.
End Misc.
Hint Unfold transpose compat_op compat_nat.
Hint Extern 1 (Setoid_Theory _ _) => constructor; congruence.
-Ltac trans_st x = match goal with
- | H Setoid_Theory _ ?eqA |- ?eqA _ _ =>
+Ltac trans_st x := match goal with
+ | H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
apply (Seq_trans _ _ H) with x; auto
end.
-Ltac sym_st = match goal with
- | H Setoid_Theory _ ?eqA |- ?eqA _ _ =>
+Ltac sym_st := match goal with
+ | H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
apply (Seq_sym _ _ H); auto
end.
-Ltac refl_st = match goal with
- | H Setoid_Theory _ ?eqA |- ?eqA _ _ =>
+Ltac refl_st := match goal with
+ | H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
apply (Seq_refl _ _ H); auto
end.
-Definition gen_st forall A : Set, Setoid_Theory _ (@eq A).
+Definition gen_st : forall A : Set, Setoid_Theory _ (@eq A).
Proof. auto. Qed.
-Module Properties (M S).
- Module ME = OrderedTypeFacts M.E.
+Module Properties (M: S).
+ Module ME := OrderedTypeFacts M.E.
Import ME.
Import M.
Import Logic. (* to unmask [eq] *)
@@ -70,106 +70,106 @@ Module Properties (M S).
(** Results about lists without duplicates *)
- Module FM = Facts M.
+ Module FM := Facts M.
Import FM.
- Definition Add (x elt) (s s' : t) :=
- forall y elt, In y s' <-> E.eq x y \/ In y s.
+ Definition Add (x : elt) (s s' : t) :=
+ forall y : elt, In y s' <-> E.eq x y \/ In y s.
- Lemma In_dec forall x s, {In x s} + {~ In x s}.
+ Lemma In_dec : forall x s, {In x s} + {~ In x s}.
Proof.
intros; generalize (mem_iff s x); case (mem x s); intuition.
Qed.
Section BasicProperties.
- Variable s s' s'' s1 s2 s3 t.
- Variable x elt.
+ Variable s s' s'' s1 s2 s3 : t.
+ Variable x : elt.
(** properties of [Equal] *)
- Lemma equal_refl s[=]s.
+ Lemma equal_refl : s[=]s.
Proof.
apply eq_refl.
Qed.
- Lemma equal_sym s[=]s' -> s'[=]s.
+ Lemma equal_sym : s[=]s' -> s'[=]s.
Proof.
apply eq_sym.
Qed.
- Lemma equal_trans s1[=]s2 -> s2[=]s3 -> s1[=]s3.
+ Lemma equal_trans : s1[=]s2 -> s2[=]s3 -> s1[=]s3.
Proof.
intros; apply eq_trans with s2; auto.
Qed.
(** properties of [Subset] *)
- Lemma subset_refl s[<=]s.
+ Lemma subset_refl : s[<=]s.
Proof.
unfold Subset; intuition.
Qed.
- Lemma subset_antisym s[<=]s' -> s'[<=]s -> s[=]s'.
+ Lemma subset_antisym : s[<=]s' -> s'[<=]s -> s[=]s'.
Proof.
unfold Subset, Equal; intuition.
Qed.
- Lemma subset_trans s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.
+ Lemma subset_trans : s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.
Proof.
unfold Subset; intuition.
Qed.
- Lemma subset_equal s[=]s' -> s[<=]s'.
+ Lemma subset_equal : s[=]s' -> s[<=]s'.
Proof.
unfold Subset, Equal; firstorder.
Qed.
- Lemma subset_empty empty[<=]s.
+ Lemma subset_empty : empty[<=]s.
Proof.
unfold Subset; intros a; set_iff; intuition.
Qed.
- Lemma subset_remove_3 s1[<=]s2 -> remove x s1 [<=] s2.
+ Lemma subset_remove_3 : s1[<=]s2 -> remove x s1 [<=] s2.
Proof.
unfold Subset; intros H a; set_iff; intuition.
Qed.
- Lemma subset_diff s1[<=]s3 -> diff s1 s2 [<=] s3.
+ Lemma subset_diff : s1[<=]s3 -> diff s1 s2 [<=] s3.
Proof.
unfold Subset; intros H a; set_iff; intuition.
Qed.
- Lemma subset_add_3 In x s2 -> s1[<=]s2 -> add x s1 [<=] s2.
+ Lemma subset_add_3 : In x s2 -> s1[<=]s2 -> add x s1 [<=] s2.
Proof.
unfold Subset; intros H H0 a; set_iff; intuition.
rewrite <- H2; auto.
Qed.
- Lemma subset_add_2 s1[<=]s2 -> s1[<=] add x s2.
+ Lemma subset_add_2 : s1[<=]s2 -> s1[<=] add x s2.
Proof.
unfold Subset; intuition.
Qed.
- Lemma in_subset In x s1 -> s1[<=]s2 -> In x s2.
+ Lemma in_subset : In x s1 -> s1[<=]s2 -> In x s2.
Proof.
unfold Subset; intuition.
Qed.
(** properties of [empty] *)
- Lemma empty_is_empty_1 Empty s -> s[=]empty.
+ Lemma empty_is_empty_1 : Empty s -> s[=]empty.
Proof.
unfold Empty, Equal; intros; generalize (H a); set_iff; tauto.
Qed.
- Lemma empty_is_empty_2 s[=]empty -> Empty s.
+ Lemma empty_is_empty_2 : s[=]empty -> Empty s.
Proof.
unfold Empty, Equal; intros; generalize (H a); set_iff; tauto.
Qed.
(** properties of [add] *)
- Lemma add_equal In x s -> add x s [=] s.
+ Lemma add_equal : In x s -> add x s [=] s.
Proof.
unfold Equal; intros; set_iff; intuition.
rewrite <- H1; auto.
@@ -177,26 +177,26 @@ Module Properties (M S).
(** properties of [remove] *)
- Lemma remove_equal ~ In x s -> remove x s [=] s.
+ Lemma remove_equal : ~ In x s -> remove x s [=] s.
Proof.
unfold Equal; intros; set_iff; intuition.
rewrite H1 in H; auto.
Qed.
- Lemma Equal_remove s[=]s' -> remove x s [=] remove x s'.
+ Lemma Equal_remove : s[=]s' -> remove x s [=] remove x s'.
Proof.
intros; rewrite H; apply eq_refl.
Qed.
(** properties of [add] and [remove] *)
- Lemma add_remove In x s -> add x (remove x s) [=] s.
+ Lemma add_remove : In x s -> add x (remove x s) [=] s.
Proof.
unfold Equal; intros; set_iff; elim (eq_dec x a); intuition.
rewrite <- H1; auto.
Qed.
- Lemma remove_add ~In x s -> remove x (add x s) [=] s.
+ Lemma remove_add : ~In x s -> remove x (add x s) [=] s.
Proof.
unfold Equal; intros; set_iff; elim (eq_dec x a); intuition.
rewrite H1 in H; auto.
@@ -204,148 +204,148 @@ Module Properties (M S).
(** properties of [singleton] *)
- Lemma singleton_equal_add singleton x [=] add x empty.
+ Lemma singleton_equal_add : singleton x [=] add x empty.
Proof.
unfold Equal; intros; set_iff; intuition.
Qed.
(** properties of [union] *)
- Lemma union_sym union s s' [=] union s' s.
+ Lemma union_sym : union s s' [=] union s' s.
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma union_subset_equal s[<=]s' -> union s s' [=] s'.
+ Lemma union_subset_equal : s[<=]s' -> union s s' [=] s'.
Proof.
unfold Subset, Equal; intros; set_iff; intuition.
Qed.
- Lemma union_equal_1 s[=]s' -> union s s'' [=] union s' s''.
+ Lemma union_equal_1 : s[=]s' -> union s s'' [=] union s' s''.
Proof.
intros; rewrite H; apply eq_refl.
Qed.
- Lemma union_equal_2 s'[=]s'' -> union s s' [=] union s s''.
+ Lemma union_equal_2 : s'[=]s'' -> union s s' [=] union s s''.
Proof.
intros; rewrite H; apply eq_refl.
Qed.
- Lemma union_assoc union (union s s') s'' [=] union s (union s' s'').
+ Lemma union_assoc : union (union s s') s'' [=] union s (union s' s'').
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma add_union_singleton add x s [=] union (singleton x) s.
+ Lemma add_union_singleton : add x s [=] union (singleton x) s.
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma union_add union (add x s) s' [=] add x (union s s').
+ Lemma union_add : union (add x s) s' [=] add x (union s s').
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma union_subset_1 s [<=] union s s'.
+ Lemma union_subset_1 : s [<=] union s s'.
Proof.
unfold Subset; intuition.
Qed.
- Lemma union_subset_2 s' [<=] union s s'.
+ Lemma union_subset_2 : s' [<=] union s s'.
Proof.
unfold Subset; intuition.
Qed.
- Lemma union_subset_3 s[<=]s'' -> s'[<=]s'' -> union s s' [<=] s''.
+ Lemma union_subset_3 : s[<=]s'' -> s'[<=]s'' -> union s s' [<=] s''.
Proof.
unfold Subset; intros H H0 a; set_iff; intuition.
Qed.
- Lemma empty_union_1 Empty s -> union s s' [=] s'.
+ Lemma empty_union_1 : Empty s -> union s s' [=] s'.
Proof.
unfold Equal, Empty; intros; set_iff; firstorder.
Qed.
- Lemma empty_union_2 Empty s -> union s' s [=] s'.
+ Lemma empty_union_2 : Empty s -> union s' s [=] s'.
Proof.
unfold Equal, Empty; intros; set_iff; firstorder.
Qed.
- Lemma not_in_union ~In x s -> ~In x s' -> ~In x (union s s').
+ Lemma not_in_union : ~In x s -> ~In x s' -> ~In x (union s s').
Proof.
intros; set_iff; intuition.
Qed.
(** properties of [inter] *)
- Lemma inter_sym inter s s' [=] inter s' s.
+ Lemma inter_sym : inter s s' [=] inter s' s.
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma inter_subset_equal s[<=]s' -> inter s s' [=] s.
+ Lemma inter_subset_equal : s[<=]s' -> inter s s' [=] s.
Proof.
unfold Equal; intros; set_iff; intuition.
Qed.
- Lemma inter_equal_1 s[=]s' -> inter s s'' [=] inter s' s''.
+ Lemma inter_equal_1 : s[=]s' -> inter s s'' [=] inter s' s''.
Proof.
intros; rewrite H; apply eq_refl.
Qed.
- Lemma inter_equal_2 s'[=]s'' -> inter s s' [=] inter s s''.
+ Lemma inter_equal_2 : s'[=]s'' -> inter s s' [=] inter s s''.
Proof.
intros; rewrite H; apply eq_refl.
Qed.
- Lemma inter_assoc inter (inter s s') s'' [=] inter s (inter s' s'').
+ Lemma inter_assoc : inter (inter s s') s'' [=] inter s (inter s' s'').
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma union_inter_1 inter (union s s') s'' [=] union (inter s s'') (inter s' s'').
+ Lemma union_inter_1 : inter (union s s') s'' [=] union (inter s s'') (inter s' s'').
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma union_inter_2 union (inter s s') s'' [=] inter (union s s'') (union s' s'').
+ Lemma union_inter_2 : union (inter s s') s'' [=] inter (union s s'') (union s' s'').
Proof.
unfold Equal; intros; set_iff; tauto.
Qed.
- Lemma inter_add_1 In x s' -> inter (add x s) s' [=] add x (inter s s').
+ Lemma inter_add_1 : In x s' -> inter (add x s) s' [=] add x (inter s s').
Proof.
unfold Equal; intros; set_iff; intuition.
rewrite <- H1; auto.
Qed.
- Lemma inter_add_2 ~ In x s' -> inter (add x s) s' [=] inter s s'.
+ Lemma inter_add_2 : ~ In x s' -> inter (add x s) s' [=] inter s s'.
Proof.
unfold Equal; intros; set_iff; intuition.
destruct H; rewrite H0; auto.
Qed.
- Lemma empty_inter_1 Empty s -> Empty (inter s s').
+ Lemma empty_inter_1 : Empty s -> Empty (inter s s').
Proof.
unfold Empty; intros; set_iff; firstorder.
Qed.
- Lemma empty_inter_2 Empty s' -> Empty (inter s s').
+ Lemma empty_inter_2 : Empty s' -> Empty (inter s s').
Proof.
unfold Empty; intros; set_iff; firstorder.
Qed.
- Lemma inter_subset_1 inter s s' [<=] s.
+ Lemma inter_subset_1 : inter s s' [<=] s.
Proof.
unfold Subset; intro a; set_iff; tauto.
Qed.
- Lemma inter_subset_2 inter s s' [<=] s'.
+ Lemma inter_subset_2 : inter s s' [<=] s'.
Proof.
unfold Subset; intro a; set_iff; tauto.
Qed.
- Lemma inter_subset_3
+ Lemma inter_subset_3 :
s''[<=]s -> s''[<=]s' -> s''[<=] inter s s'.
Proof.
unfold Subset; intros H H' a; set_iff; intuition.
@@ -353,38 +353,38 @@ Module Properties (M S).
(** properties of [diff] *)
- Lemma empty_diff_1 Empty s -> Empty (diff s s').
+ Lemma empty_diff_1 : Empty s -> Empty (diff s s').
Proof.
unfold Empty, Equal; intros; set_iff; firstorder.
Qed.
- Lemma empty_diff_2 Empty s -> diff s' s [=] s'.
+ Lemma empty_diff_2 : Empty s -> diff s' s [=] s'.
Proof.
unfold Empty, Equal; intros; set_iff; firstorder.
Qed.
- Lemma diff_subset diff s s' [<=] s.
+ Lemma diff_subset : diff s s' [<=] s.
Proof.
unfold Subset; intros a; set_iff; tauto.
Qed.
- Lemma diff_subset_equal s[<=]s' -> diff s s' [=] empty.
+ Lemma diff_subset_equal : s[<=]s' -> diff s s' [=] empty.
Proof.
unfold Subset, Equal; intros; set_iff; intuition; absurd (In a empty); auto.
Qed.
- Lemma remove_diff_singleton
+ Lemma remove_diff_singleton :
remove x s [=] diff s (singleton x).
Proof.
unfold Equal; intros; set_iff; intuition.
Qed.
- Lemma diff_inter_empty inter (diff s s') (inter s s') [=] empty.
+ Lemma diff_inter_empty : inter (diff s s') (inter s s') [=] empty.
Proof.
unfold Equal; intros; set_iff; intuition; absurd (In a empty); auto.
Qed.
- Lemma diff_inter_all union (diff s s') (inter s s') [=] s.
+ Lemma diff_inter_all : union (diff s s') (inter s s') [=] s.
Proof.
unfold Equal; intros; set_iff; intuition.
elim (In_dec a s'); auto.
@@ -392,38 +392,38 @@ Module Properties (M S).
(** properties of [Add] *)
- Lemma Add_add Add x s (add x s).
+ Lemma Add_add : Add x s (add x s).
Proof.
unfold Add; intros; set_iff; intuition.
Qed.
- Lemma Add_remove In x s -> Add x (remove x s) s.
+ Lemma Add_remove : In x s -> Add x (remove x s) s.
Proof.
unfold Add; intros; set_iff; intuition.
elim (eq_dec x y); auto.
rewrite <- H1; auto.
Qed.
- Lemma union_Add Add x s s' -> Add x (union s s'') (union s' s'').
+ Lemma union_Add : Add x s s' -> Add x (union s s'') (union s' s'').
Proof.
unfold Add; intros; set_iff; rewrite H; tauto.
Qed.
- Lemma inter_Add
+ Lemma inter_Add :
In x s'' -> Add x s s' -> Add x (inter s s'') (inter s' s'').
Proof.
unfold Add; intros; set_iff; rewrite H0; intuition.
rewrite <- H2; auto.
Qed.
- Lemma union_Equal
+ Lemma union_Equal :
In x s'' -> Add x s s' -> union s s'' [=] union s' s''.
Proof.
unfold Add, Equal; intros; set_iff; rewrite H0; intuition.
rewrite <- H1; auto.
Qed.
- Lemma inter_Add_2
+ Lemma inter_Add_2 :
~In x s'' -> Add x s s' -> inter s s'' [=] inter s' s''.
Proof.
unfold Add, Equal; intros; set_iff; rewrite H0; intuition.
@@ -432,10 +432,10 @@ Module Properties (M S).
End BasicProperties.
- Hint Immediate equal_sym set.
- Hint Resolve equal_refl equal_trans set.
+ Hint Immediate equal_sym: set.
+ Hint Resolve equal_refl equal_trans : set.
- Hint Immediate add_remove remove_add union_sym inter_sym set.
+ Hint Immediate add_remove remove_add union_sym inter_sym: set.
Hint Resolve subset_refl subset_equal subset_antisym
subset_trans subset_empty subset_remove_3 subset_diff subset_add_3
subset_add_2 in_subset empty_is_empty_1 empty_is_empty_2 add_equal
@@ -447,18 +447,18 @@ Module Properties (M S).
empty_diff_2 union_Add inter_Add union_Equal inter_Add_2 not_in_union
inter_subset_1 inter_subset_2 inter_subset_3 diff_subset diff_subset_equal
remove_diff_singleton diff_inter_empty diff_inter_all Add_add Add_remove
- Equal_remove set.
+ Equal_remove : set.
- Notation NoRedun = (noredunA E.eq).
+ Notation NoRedun := (noredunA E.eq).
Section noredunA_Remove.
- Definition remove_list x l = MoreList.filter (fun y => negb (eqb x y)) l.
+ Definition remove_list x l := MoreList.filter (fun y => negb (eqb x y)) l.
- Lemma remove_list_correct
+ Lemma remove_list_correct :
forall s x, NoRedun s ->
NoRedun (remove_list x s) /\
- (forall y elt, ME.In y (remove_list x s) <-> ME.In y s /\ ~ E.eq x y).
+ (forall y : elt, ME.In y (remove_list x s) <-> ME.In y s /\ ~ E.eq x y).
Proof.
simple induction s; simpl; intros.
repeat (split; trivial).
@@ -483,11 +483,11 @@ Module Properties (M S).
constructor 2; rewrite H4; auto.
Qed.
- Let ListEq l l' = forall y : elt, ME.In y l <-> ME.In y l'.
+ Let ListEq l l' := forall y : elt, ME.In y l <-> ME.In y l'.
- Lemma remove_list_equal
- forall s s' x, NoRedun (x : s) -> NoRedun s' ->
- ListEq (x : s) s' -> ListEq s (remove_list x s').
+ Lemma remove_list_equal :
+ forall s s' x, NoRedun (x :: s) -> NoRedun s' ->
+ ListEq (x :: s) s' -> ListEq s (remove_list x s').
Proof.
unfold ListEq; intros.
inversion_clear H.
@@ -502,12 +502,12 @@ Module Properties (M S).
destruct H9; auto.
Qed.
- Let ListAdd x l l' = forall y : elt, ME.In y l' <-> E.eq x y \/ ME.In y l.
+ Let ListAdd x l l' := forall y : elt, ME.In y l' <-> E.eq x y \/ ME.In y l.
- Lemma remove_list_add
- forall s s' x x', NoRedun s -> NoRedun (x' : s') ->
+ Lemma remove_list_add :
+ forall s s' x x', NoRedun s -> NoRedun (x' :: s') ->
~ E.eq x x' -> ~ ME.In x s ->
- ListAdd x s (x' : s') -> ListAdd x (remove_list x' s) s'.
+ ListAdd x s (x' :: s') -> ListAdd x (remove_list x' s) s'.
Proof.
unfold ListAdd; intros.
inversion_clear H0.
@@ -520,20 +520,20 @@ Module Properties (M S).
right; apply H8; split; auto.
swap H4; apply In_eq with y; auto.
destruct H3.
- assert (ME.In y (x' : s')). auto.
+ assert (ME.In y (x' :: s')). auto.
inversion_clear H10; auto.
destruct H1; order.
destruct (H7 H3).
- assert (ME.In y (x' : s')). auto.
+ assert (ME.In y (x' :: s')). auto.
inversion_clear H12; auto.
destruct H11; auto.
Qed.
- Variables (ASet)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).
- Variables (felt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).
- Variables (iA).
+ Variables (A:Set)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).
+ Variables (f:elt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).
+ Variables (i:A).
- Lemma remove_list_fold_right_0
+ Lemma remove_list_fold_right_0 :
forall s x, NoRedun s -> ~ME.In x s ->
eqA (fold_right f i s) (fold_right f i (remove_list x s)).
Proof.
@@ -545,7 +545,7 @@ Module Properties (M S).
apply Comp; auto.
Qed.
- Lemma remove_list_fold_right
+ Lemma remove_list_fold_right :
forall s x, NoRedun s -> ME.In x s ->
eqA (fold_right f i s) (f x (fold_right f i (remove_list x s))).
Proof.
@@ -562,7 +562,7 @@ Module Properties (M S).
trans_st (f a (f x (fold_right f i (remove_list x l)))).
Qed.
- Lemma fold_right_equal
+ Lemma fold_right_equal :
forall s s', NoRedun s -> NoRedun s' ->
ListEq s s' -> eqA (fold_right f i s) (fold_right f i s').
Proof.
@@ -571,7 +571,7 @@ Module Properties (M S).
intros; refl_st; auto.
unfold ListEq; intros.
destruct (H1 t0).
- assert (X ME.In t0 nil); auto; inversion X.
+ assert (X : ME.In t0 nil); auto; inversion X.
intros x l Hrec s' N N' E; simpl in *.
trans_st (f x (fold_right f i (remove_list x s'))).
apply Comp; auto.
@@ -585,14 +585,14 @@ Module Properties (M S).
rewrite <- E; auto.
Qed.
- Lemma fold_right_add
+ Lemma fold_right_add :
forall s' s x, NoRedun s -> NoRedun s' -> ~ ME.In x s ->
ListAdd x s s' -> eqA (fold_right f i s') (f x (fold_right f i s)).
Proof.
simple induction s'.
unfold ListAdd; intros.
destruct (H2 x); clear H2.
- assert (X ME.In x nil); auto; inversion X.
+ assert (X : ME.In x nil); auto; inversion X.
intros x' l' Hrec s x N N' IN EQ; simpl.
(* if x=x' *)
destruct (eq_dec x x').
@@ -605,7 +605,7 @@ Module Properties (M S).
destruct H; auto.
inversion_clear N'.
destruct H2; apply In_eq with y; auto; order.
- assert (XME.In y (x' :: l')); auto; inversion_clear X; auto.
+ assert (X:ME.In y (x' :: l')); auto; inversion_clear X; auto.
destruct IN; apply In_eq with y; auto; order.
(* else x<>x' *)
trans_st (f x' (f x (fold_right f i (remove_list x' s)))).
@@ -632,14 +632,14 @@ Module Properties (M S).
Section Old_Spec_Now_Properties.
(** When [FSets] was first designed, the order in which Ocaml's [Set.fold]
- takes the set elements was unspecified. This specification reflects this fact
+ takes the set elements was unspecified. This specification reflects this fact:
*)
- Lemma fold_0
- forall s (A Set) (i : A) (f : elt -> A -> A),
- exists l list elt,
+ Lemma fold_0 :
+ forall s (A : Set) (i : A) (f : elt -> A -> A),
+ exists l : list elt,
NoRedun l /\
- (forall x elt, In x s <-> InA E.eq x l) /\
+ (forall x : elt, In x s <-> InA E.eq x l) /\
fold f s i = fold_right f i l.
Proof.
intros; exists (rev (elements s)); split.
@@ -656,9 +656,9 @@ Module Properties (M S).
the recursive structure of a set. It is now lemmas [fold_1] and
[fold_2]. *)
- Lemma fold_1
- forall s (A Set) (eqA : A -> A -> Prop)
- (st Setoid_Theory A eqA) (i : A) (f : elt -> A -> A),
+ Lemma fold_1 :
+ forall s (A : Set) (eqA : A -> A -> Prop)
+ (st : Setoid_Theory A eqA) (i : A) (f : elt -> A -> A),
Empty s -> eqA (fold f s i) i.
Proof.
unfold Empty; intros; destruct (fold_0 s i f) as (l,(H1, (H2, H3))).
@@ -669,9 +669,9 @@ Module Properties (M S).
elim (H2 e); intuition.
Qed.
- Lemma fold_2
- forall s s' x (A Set) (eqA : A -> A -> Prop)
- (st Setoid_Theory A eqA) (i : A) (f : elt -> A -> A),
+ Lemma fold_2 :
+ forall s s' x (A : Set) (eqA : A -> A -> Prop)
+ (st : Setoid_Theory A eqA) (i : A) (f : elt -> A -> A),
compat_op E.eq eqA f ->
transpose eqA f ->
~ In x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).
@@ -679,34 +679,34 @@ Module Properties (M S).
intros; destruct (fold_0 s i f) as (l,(Hl, (Hl1, Hl2)));
destruct (fold_0 s' i f) as (l',(Hl', (Hl'1, Hl'2))).
rewrite Hl2; rewrite Hl'2; clear Hl2 Hl'2.
- apply fold_right_add with (eqA = eqA); auto.
+ apply fold_right_add with (eqA := eqA); auto.
rewrite <- Hl1; auto.
intros; rewrite <- Hl1; rewrite <- Hl'1; auto.
Qed.
(** Similar specifications for [cardinal]. *)
- Lemma cardinal_fold forall s, cardinal s = fold (fun _ => S) s 0.
+ Lemma cardinal_fold : forall s, cardinal s = fold (fun _ => S) s 0.
Proof.
intros; rewrite cardinal_1; rewrite M.fold_1.
symmetry; apply fold_left_length; auto.
Qed.
- Lemma cardinal_0
- forall s, exists l list elt,
+ Lemma cardinal_0 :
+ forall s, exists l : list elt,
noredunA E.eq l /\
- (forall x elt, In x s <-> InA E.eq x l) /\
+ (forall x : elt, In x s <-> InA E.eq x l) /\
cardinal s = length l.
Proof.
intros; exists (elements s); intuition; apply cardinal_1.
Qed.
- Lemma cardinal_1 forall s, Empty s -> cardinal s = 0.
+ Lemma cardinal_1 : forall s, Empty s -> cardinal s = 0.
Proof.
intros; rewrite cardinal_fold; apply fold_1; auto.
Qed.
- Lemma cardinal_2
+ Lemma cardinal_2 :
forall s s' x, ~ In x s -> Add x s s' -> cardinal s' = S (cardinal s).
Proof.
intros; do 2 rewrite cardinal_fold.
@@ -718,7 +718,7 @@ Module Properties (M S).
(** * Induction principle over sets *)
- Lemma cardinal_inv_1 forall s, cardinal s = 0 -> Empty s.
+ Lemma cardinal_inv_1 : forall s, cardinal s = 0 -> Empty s.
Proof.
intros s; rewrite M.cardinal_1; intros H a; red.
rewrite elements_iff.
@@ -726,16 +726,16 @@ Module Properties (M S).
Qed.
Hint Resolve cardinal_inv_1.
- Lemma cardinal_inv_2
- forall s n, cardinal s = S n -> { x elt | In x s }.
+ Lemma cardinal_inv_2 :
+ forall s n, cardinal s = S n -> { x : elt | In x s }.
Proof.
intros; rewrite M.cardinal_1 in H.
- generalize (elements_2 (s=s)).
+ generalize (elements_2 (s:=s)).
destruct (elements s); try discriminate.
exists e; auto.
Qed.
- Lemma Equal_cardinal_aux
+ Lemma Equal_cardinal_aux :
forall n s s', cardinal s = n -> s[=]s' -> cardinal s = cardinal s'.
Proof.
simple induction n; intros.
@@ -744,25 +744,25 @@ Module Properties (M S).
rewrite <- H0; auto.
destruct (cardinal_inv_2 H0) as (x,H2).
revert H0.
- rewrite (cardinal_2 (s=remove x s) (s':=s) (x:=x)); auto with set.
- rewrite (cardinal_2 (s=remove x s') (s':=s') (x:=x)); auto with set.
+ rewrite (cardinal_2 (s:=remove x s) (s':=s) (x:=x)); auto with set.
+ rewrite (cardinal_2 (s:=remove x s') (s':=s') (x:=x)); auto with set.
rewrite H1 in H2; auto with set.
Qed.
- Lemma Equal_cardinal forall s s', s[=]s' -> cardinal s = cardinal s'.
+ Lemma Equal_cardinal : forall s s', s[=]s' -> cardinal s = cardinal s'.
Proof.
intros; apply Equal_cardinal_aux with (cardinal s); auto.
Qed.
- Add Morphism cardinal cardinal_m.
+ Add Morphism cardinal : cardinal_m.
Proof.
exact Equal_cardinal.
Qed.
Hint Resolve Add_add Add_remove Equal_remove cardinal_inv_1 Equal_cardinal.
- Lemma cardinal_induction
- forall P t -> Type,
+ Lemma cardinal_induction :
+ forall P : t -> Type,
(forall s, Empty s -> P s) ->
(forall s s', P s -> forall x, ~In x s -> Add x s s' -> P s') ->
forall n s, cardinal s = n -> P s.
@@ -771,14 +771,14 @@ Module Properties (M S).
destruct (cardinal_inv_2 H) as (x,H0).
apply X0 with (remove x s) x; auto.
apply X1; auto.
- rewrite (cardinal_2 (x=x)(s:=remove x s)(s':=s)) in H; auto.
+ rewrite (cardinal_2 (x:=x)(s:=remove x s)(s':=s)) in H; auto.
Qed.
- Lemma set_induction
- forall P t -> Type,
- (forall s t, Empty s -> P s) ->
- (forall s s' t, P s -> forall x : elt, ~In x s -> Add x s s' -> P s') ->
- forall s t, P s.
+ Lemma set_induction :
+ forall P : t -> Type,
+ (forall s : t, Empty s -> P s) ->
+ (forall s s' : t, P s -> forall x : elt, ~In x s -> Add x s s' -> P s') ->
+ forall s : t, P s.
Proof.
intros; apply cardinal_induction with (cardinal s); auto.
Qed.
@@ -786,18 +786,18 @@ Module Properties (M S).
(** Other properties of [fold]. *)
Section Fold.
- Variables (ASet)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).
- Variables (felt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).
+ Variables (A:Set)(eqA:A->A->Prop)(st:Setoid_Theory _ eqA).
+ Variables (f:elt->A->A)(Comp:compat_op E.eq eqA f)(Ass:transpose eqA f).
Section Fold_1.
- Variable i i'A.
+ Variable i i':A.
- Lemma fold_empty eqA (fold f empty i) i.
+ Lemma fold_empty : eqA (fold f empty i) i.
Proof.
apply fold_1; auto.
Qed.
- Lemma fold_equal
+ Lemma fold_equal :
forall s s', s[=]s' -> eqA (fold f s i) (fold f s' i).
Proof.
intros s; pattern s; apply set_induction; clear s; intros.
@@ -806,40 +806,40 @@ Module Properties (M S).
sym_st; apply fold_1; auto.
rewrite <- H0; auto.
trans_st (f x (fold f s i)).
- apply fold_2 with (eqA = eqA); auto.
- sym_st; apply fold_2 with (eqA = eqA); auto.
+ apply fold_2 with (eqA := eqA); auto.
+ sym_st; apply fold_2 with (eqA := eqA); auto.
unfold Add in *; intros.
rewrite <- H2; auto.
Qed.
- Lemma fold_add forall s x, ~In x s ->
+ Lemma fold_add : forall s x, ~In x s ->
eqA (fold f (add x s) i) (f x (fold f s i)).
Proof.
- intros; apply fold_2 with (eqA = eqA); auto.
+ intros; apply fold_2 with (eqA := eqA); auto.
Qed.
- Lemma add_fold forall s x, In x s ->
+ Lemma add_fold : forall s x, In x s ->
eqA (fold f (add x s) i) (fold f s i).
Proof.
intros; apply fold_equal; auto with set.
Qed.
- Lemma remove_fold_1 forall s x, In x s ->
+ Lemma remove_fold_1: forall s x, In x s ->
eqA (f x (fold f (remove x s) i)) (fold f s i).
Proof.
intros.
sym_st.
- apply fold_2 with (eqA=eqA); auto.
+ apply fold_2 with (eqA:=eqA); auto.
Qed.
- Lemma remove_fold_2 forall s x, ~In x s ->
+ Lemma remove_fold_2: forall s x, ~In x s ->
eqA (fold f (remove x s) i) (fold f s i).
Proof.
intros.
apply fold_equal; auto with set.
Qed.
- Lemma fold_commutes forall s x,
+ Lemma fold_commutes : forall s x,
eqA (fold f s (f x i)) (f x (fold f s i)).
Proof.
intros; pattern s; apply set_induction; clear s; intros.
@@ -849,15 +849,15 @@ Module Properties (M S).
apply Comp; auto.
apply fold_1; auto.
trans_st (f x0 (fold f s (f x i))).
- apply fold_2 with (eqA=eqA); auto.
+ apply fold_2 with (eqA:=eqA); auto.
trans_st (f x0 (f x (fold f s i))).
trans_st (f x (f x0 (fold f s i))).
apply Comp; auto.
sym_st.
- apply fold_2 with (eqA=eqA); auto.
+ apply fold_2 with (eqA:=eqA); auto.
Qed.
- Lemma fold_init forall s, eqA i i' ->
+ Lemma fold_init : forall s, eqA i i' ->
eqA (fold f s i) (fold f s i').
Proof.
intros; pattern s; apply set_induction; clear s; intros.
@@ -866,16 +866,16 @@ Module Properties (M S).
trans_st i'.
sym_st; apply fold_1; auto.
trans_st (f x (fold f s i)).
- apply fold_2 with (eqA=eqA); auto.
+ apply fold_2 with (eqA:=eqA); auto.
trans_st (f x (fold f s i')).
- sym_st; apply fold_2 with (eqA=eqA); auto.
+ sym_st; apply fold_2 with (eqA:=eqA); auto.
Qed.
End Fold_1.
Section Fold_2.
- Variable iA.
+ Variable i:A.
- Lemma fold_union_inter forall s s',
+ Lemma fold_union_inter : forall s s',
eqA (fold f (union s s') (fold f (inter s s') i))
(fold f s (fold f s' i)).
Proof.
@@ -891,7 +891,7 @@ Module Properties (M S).
(* In x s' *)
trans_st (fold f (union s'' s') (f x (fold f (inter s s') i))); auto with set.
apply fold_init; auto.
- apply fold_2 with (eqA=eqA); auto with set.
+ apply fold_2 with (eqA:=eqA); auto with set.
rewrite inter_iff; intuition.
trans_st (f x (fold f s (fold f s' i))).
trans_st (fold f (union s s') (f x (fold f (inter s s') i))).
@@ -899,24 +899,24 @@ Module Properties (M S).
apply equal_sym; apply union_Equal with x; auto with set.
trans_st (f x (fold f (union s s') (fold f (inter s s') i))).
apply fold_commutes; auto.
- sym_st; apply fold_2 with (eqA=eqA); auto.
+ sym_st; apply fold_2 with (eqA:=eqA); auto.
(* ~(In x s') *)
trans_st (f x (fold f (union s s') (fold f (inter s'' s') i))).
- apply fold_2 with (eqA=eqA); auto with set.
+ apply fold_2 with (eqA:=eqA); auto with set.
trans_st (f x (fold f (union s s') (fold f (inter s s') i))).
apply Comp;auto.
apply fold_init;auto.
apply fold_equal;auto.
apply equal_sym; apply inter_Add_2 with x; auto with set.
trans_st (f x (fold f s (fold f s' i))).
- sym_st; apply fold_2 with (eqA=eqA); auto.
+ sym_st; apply fold_2 with (eqA:=eqA); auto.
Qed.
End Fold_2.
Section Fold_3.
- Variable iA.
+ Variable i:A.
- Lemma fold_diff_inter forall s s',
+ Lemma fold_diff_inter : forall s s',
eqA (fold f (diff s s') (fold f (inter s s') i)) (fold f s i).
Proof.
intros.
@@ -929,7 +929,7 @@ Module Properties (M S).
apply fold_1; auto with set.
Qed.
- Lemma fold_union forall s s', (forall x, ~In x s\/~In x s') ->
+ Lemma fold_union: forall s s', (forall x, ~In x s\/~In x s') ->
eqA (fold f (union s s') i) (fold f s (fold f s' i)).
Proof.
intros.
@@ -943,12 +943,12 @@ Module Properties (M S).
End Fold_3.
End Fold.
- Lemma fold_plus
+ Lemma fold_plus :
forall s p, fold (fun _ => S) s p = fold (fun _ => S) s 0 + p.
Proof.
- assert (st = gen_st nat).
- assert (fe compat_op E.eq (@eq _) (fun _ => S)) by unfold compat_op; auto.
- assert (fp transpose (@eq _) (fun _:elt => S)) by unfold transpose; auto.
+ assert (st := gen_st nat).
+ assert (fe : compat_op E.eq (@eq _) (fun _ => S)) by unfold compat_op; auto.
+ assert (fp : transpose (@eq _) (fun _:elt => S)) by unfold transpose; auto.
intros s p; pattern s; apply set_induction; clear s; intros.
rewrite (fold_1 st p (fun _ => S) H).
rewrite (fold_1 st 0 (fun _ => S) H); trivial.
@@ -963,14 +963,14 @@ Module Properties (M S).
(** properties of [cardinal] *)
- Lemma empty_cardinal cardinal empty = 0.
+ Lemma empty_cardinal : cardinal empty = 0.
Proof.
rewrite cardinal_fold; apply fold_1; auto.
Qed.
- Hint Immediate empty_cardinal cardinal_1 set.
+ Hint Immediate empty_cardinal cardinal_1 : set.
- Lemma singleton_cardinal forall x, cardinal (singleton x) = 1.
+ Lemma singleton_cardinal : forall x, cardinal (singleton x) = 1.
Proof.
intros.
rewrite (singleton_equal_add x).
@@ -978,17 +978,17 @@ Module Properties (M S).
apply cardinal_2 with x; auto with set.
Qed.
- Hint Resolve singleton_cardinal set.
+ Hint Resolve singleton_cardinal: set.
- Lemma diff_inter_cardinal
+ Lemma diff_inter_cardinal :
forall s s', cardinal (diff s s') + cardinal (inter s s') = cardinal s .
Proof.
intros; do 3 rewrite cardinal_fold.
rewrite <- fold_plus.
- apply fold_diff_inter with (eqA=@eq nat); auto.
+ apply fold_diff_inter with (eqA:=@eq nat); auto.
Qed.
- Lemma union_cardinal
+ Lemma union_cardinal:
forall s s', (forall x, ~In x s\/~In x s') ->
cardinal (union s s')=cardinal s+cardinal s'.
Proof.
@@ -997,7 +997,7 @@ Module Properties (M S).
apply fold_union; auto.
Qed.
- Lemma subset_cardinal
+ Lemma subset_cardinal :
forall s s', s[<=]s' -> cardinal s <= cardinal s' .
Proof.
intros.
@@ -1006,47 +1006,47 @@ Module Properties (M S).
rewrite (inter_subset_equal H); auto with arith.
Qed.
- Lemma union_inter_cardinal
+ Lemma union_inter_cardinal :
forall s s', cardinal (union s s') + cardinal (inter s s') = cardinal s + cardinal s' .
Proof.
intros.
do 4 rewrite cardinal_fold.
do 2 rewrite <- fold_plus.
- apply fold_union_inter with (eqA=@eq nat); auto.
+ apply fold_union_inter with (eqA:=@eq nat); auto.
Qed.
- Lemma union_cardinal_le
+ Lemma union_cardinal_le :
forall s s', cardinal (union s s') <= cardinal s + cardinal s'.
Proof.
intros; generalize (union_inter_cardinal s s').
intros; rewrite <- H; auto with arith.
Qed.
- Lemma add_cardinal_1
+ Lemma add_cardinal_1 :
forall s x, In x s -> cardinal (add x s) = cardinal s.
Proof.
auto with set.
Qed.
- Lemma add_cardinal_2
+ Lemma add_cardinal_2 :
forall s x, ~In x s -> cardinal (add x s) = S (cardinal s).
Proof.
intros.
do 2 rewrite cardinal_fold.
change S with ((fun _ => S) x);
- apply fold_add with (eqA=@eq nat); auto.
+ apply fold_add with (eqA:=@eq nat); auto.
Qed.
- Lemma remove_cardinal_1
+ Lemma remove_cardinal_1 :
forall s x, In x s -> S (cardinal (remove x s)) = cardinal s.
Proof.
intros.
do 2 rewrite cardinal_fold.
change S with ((fun _ =>S) x).
- apply remove_fold_1 with (eqA=@eq nat); auto.
+ apply remove_fold_1 with (eqA:=@eq nat); auto.
Qed.
- Lemma remove_cardinal_2
+ Lemma remove_cardinal_2 :
forall s x, ~In x s -> cardinal (remove x s) = cardinal s.
Proof.
auto with set.
diff --git a/theories/FSets/FSetWeak.v b/theories/FSets/FSetWeak.v
index 33c06dd6d..5fea8b6ea 100644
--- a/theories/FSets/FSetWeak.v
+++ b/theories/FSets/FSetWeak.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSet.v,v 1.2 2004/12/08 19:19:24 letouzey Exp $ *)
Require Export DecidableType.
Require Export FSetWeakInterface.
diff --git a/theories/FSets/FSetWeakInterface.v b/theories/FSets/FSetWeakInterface.v
index cec0a901f..adfa5f453 100644
--- a/theories/FSets/FSetWeakInterface.v
+++ b/theories/FSets/FSetWeakInterface.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetWeakInterface.v,v 1.4 2006/02/27 15:39:44 letouzey Exp $ *)
(** * Finite sets library *)
@@ -18,12 +18,12 @@ Set Implicit Arguments.
Unset Strict Implicit.
(** Compatibility of a boolean function with respect to an equality. *)
-Definition compat_bool (ASet)(eqA: A->A->Prop)(f: A-> bool) :=
- forall x y A, eqA x y -> f x = f y.
+Definition compat_bool (A:Set)(eqA: A->A->Prop)(f: A-> bool) :=
+ forall x y : A, eqA x y -> f x = f y.
(** Compatibility of a predicate with respect to an equality. *)
-Definition compat_P (ASet)(eqA: A->A->Prop)(P : A -> Prop) :=
- forall x y A, eqA x y -> P x -> P y.
+Definition compat_P (A:Set)(eqA: A->A->Prop)(P : A -> Prop) :=
+ forall x y : A, eqA x y -> P x -> P y.
Hint Unfold compat_bool compat_P.
@@ -34,201 +34,201 @@ Hint Unfold compat_bool compat_P.
Module Type S.
- Declare Module E DecidableType.
- Definition elt = E.t.
+ Declare Module E : DecidableType.
+ Definition elt := E.t.
- Parameter t Set. (** the abstract type of sets *)
+ Parameter t : Set. (** the abstract type of sets *)
- Parameter empty t.
+ Parameter empty : t.
(** The empty set. *)
- Parameter is_empty t -> bool.
+ Parameter is_empty : t -> bool.
(** Test whether a set is empty or not. *)
- Parameter mem elt -> t -> bool.
+ Parameter mem : elt -> t -> bool.
(** [mem x s] tests whether [x] belongs to the set [s]. *)
- Parameter add elt -> t -> t.
+ Parameter add : elt -> t -> t.
(** [add x s] returns a set containing all elements of [s],
plus [x]. If [x] was already in [s], [s] is returned unchanged. *)
- Parameter singleton elt -> t.
+ Parameter singleton : elt -> t.
(** [singleton x] returns the one-element set containing only [x]. *)
- Parameter remove elt -> t -> t.
+ Parameter remove : elt -> t -> t.
(** [remove x s] returns a set containing all elements of [s],
except [x]. If [x] was not in [s], [s] is returned unchanged. *)
- Parameter union t -> t -> t.
+ Parameter union : t -> t -> t.
(** Set union. *)
- Parameter inter t -> t -> t.
+ Parameter inter : t -> t -> t.
(** Set intersection. *)
- Parameter diff t -> t -> t.
+ Parameter diff : t -> t -> t.
(** Set difference. *)
- Parameter equal t -> t -> bool.
+ Parameter equal : t -> t -> bool.
(** [equal s1 s2] tests whether the sets [s1] and [s2] are
equal, that is, contain equal elements. *)
- Parameter subset t -> t -> bool.
+ Parameter subset : t -> t -> bool.
(** [subset s1 s2] tests whether the set [s1] is a subset of
the set [s2]. *)
- (** Coq comment [iter] is useless in a purely functional world *)
- (** iter (elt -> unit) -> set -> unit. i*)
+ (** Coq comment: [iter] is useless in a purely functional world *)
+ (** iter: (elt -> unit) -> set -> unit. i*)
(** [iter f s] applies [f] in turn to all elements of [s].
The order in which the elements of [s] are presented to [f]
is unspecified. *)
- Parameter fold forall A : Set, (elt -> A -> A) -> t -> A -> A.
+ Parameter fold : forall A : Set, (elt -> A -> A) -> t -> A -> A.
(** [fold f s a] computes [(f xN ... (f x2 (f x1 a))...)],
where [x1 ... xN] are the elements of [s].
The order in which elements of [s] are presented to [f] is
unspecified. *)
- Parameter for_all (elt -> bool) -> t -> bool.
+ Parameter for_all : (elt -> bool) -> t -> bool.
(** [for_all p s] checks if all elements of the set
satisfy the predicate [p]. *)
- Parameter exists_ (elt -> bool) -> t -> bool.
+ Parameter exists_ : (elt -> bool) -> t -> bool.
(** [exists p s] checks if at least one element of
the set satisfies the predicate [p]. *)
- Parameter filter (elt -> bool) -> t -> t.
+ Parameter filter : (elt -> bool) -> t -> t.
(** [filter p s] returns the set of all elements in [s]
that satisfy predicate [p]. *)
- Parameter partition (elt -> bool) -> t -> t * t.
+ Parameter partition : (elt -> bool) -> t -> t * t.
(** [partition p s] returns a pair of sets [(s1, s2)], where
[s1] is the set of all the elements of [s] that satisfy the
predicate [p], and [s2] is the set of all the elements of
[s] that do not satisfy [p]. *)
- Parameter cardinal t -> nat.
+ Parameter cardinal : t -> nat.
(** Return the number of elements of a set. *)
- (** Coq comment nat instead of int ... *)
+ (** Coq comment: nat instead of int ... *)
- Parameter elements t -> list elt.
+ Parameter elements : t -> list elt.
(** Return the list of all elements of the given set, in any order. *)
- Parameter choose t -> option elt.
+ Parameter choose : t -> option elt.
(** Return one element of the given set, or raise [Not_found] if
the set is empty. Which element is chosen is unspecified.
Equal sets could return different elements. *)
- (** Coq comment [Not_found] is represented by the option type *)
+ (** Coq comment: [Not_found] is represented by the option type *)
Section Spec.
- Variable s s' s'' t.
- Variable x y z elt.
+ Variable s s' s'' : t.
+ Variable x y z : elt.
- Parameter In elt -> t -> Prop.
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) s := forall x, In x s -> P x.
- Definition Exists (P elt -> Prop) s := exists x, In x s /\ P x.
+ Parameter In : elt -> t -> Prop.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
+ Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.
(** Specification of [In] *)
- Parameter In_1 E.eq x y -> In x s -> In y s.
+ Parameter In_1 : E.eq x y -> In x s -> In y s.
(** Specification of [mem] *)
- Parameter mem_1 In x s -> mem x s = true.
- Parameter mem_2 mem x s = true -> In x s.
+ Parameter mem_1 : In x s -> mem x s = true.
+ Parameter mem_2 : mem x s = true -> In x s.
(** Specification of [equal] *)
- Parameter equal_1 Equal s s' -> equal s s' = true.
- Parameter equal_2 equal s s' = true -> Equal s s'.
+ Parameter equal_1 : Equal s s' -> equal s s' = true.
+ Parameter equal_2 : equal s s' = true -> Equal s s'.
(** Specification of [subset] *)
- Parameter subset_1 Subset s s' -> subset s s' = true.
- Parameter subset_2 subset s s' = true -> Subset s s'.
+ Parameter subset_1 : Subset s s' -> subset s s' = true.
+ Parameter subset_2 : subset s s' = true -> Subset s s'.
(** Specification of [empty] *)
- Parameter empty_1 Empty empty.
+ Parameter empty_1 : Empty empty.
(** Specification of [is_empty] *)
- Parameter is_empty_1 Empty s -> is_empty s = true.
- Parameter is_empty_2 is_empty s = true -> Empty s.
+ Parameter is_empty_1 : Empty s -> is_empty s = true.
+ Parameter is_empty_2 : is_empty s = true -> Empty s.
(** Specification of [add] *)
- Parameter add_1 E.eq x y -> In y (add x s).
- Parameter add_2 In y s -> In y (add x s).
- Parameter add_3 ~ E.eq x y -> In y (add x s) -> In y s.
+ Parameter add_1 : E.eq x y -> In y (add x s).
+ Parameter add_2 : In y s -> In y (add x s).
+ Parameter add_3 : ~ E.eq x y -> In y (add x s) -> In y s.
(** Specification of [remove] *)
- Parameter remove_1 E.eq x y -> ~ In y (remove x s).
- Parameter remove_2 ~ E.eq x y -> In y s -> In y (remove x s).
- Parameter remove_3 In y (remove x s) -> In y s.
+ Parameter remove_1 : E.eq x y -> ~ In y (remove x s).
+ Parameter remove_2 : ~ E.eq x y -> In y s -> In y (remove x s).
+ Parameter remove_3 : In y (remove x s) -> In y s.
(** Specification of [singleton] *)
- Parameter singleton_1 In y (singleton x) -> E.eq x y.
- Parameter singleton_2 E.eq x y -> In y (singleton x).
+ Parameter singleton_1 : In y (singleton x) -> E.eq x y.
+ Parameter singleton_2 : E.eq x y -> In y (singleton x).
(** Specification of [union] *)
- Parameter union_1 In x (union s s') -> In x s \/ In x s'.
- Parameter union_2 In x s -> In x (union s s').
- Parameter union_3 In x s' -> In x (union s s').
+ Parameter union_1 : In x (union s s') -> In x s \/ In x s'.
+ Parameter union_2 : In x s -> In x (union s s').
+ Parameter union_3 : In x s' -> In x (union s s').
(** Specification of [inter] *)
- Parameter inter_1 In x (inter s s') -> In x s.
- Parameter inter_2 In x (inter s s') -> In x s'.
- Parameter inter_3 In x s -> In x s' -> In x (inter s s').
+ Parameter inter_1 : In x (inter s s') -> In x s.
+ Parameter inter_2 : In x (inter s s') -> In x s'.
+ Parameter inter_3 : In x s -> In x s' -> In x (inter s s').
(** Specification of [diff] *)
- Parameter diff_1 In x (diff s s') -> In x s.
- Parameter diff_2 In x (diff s s') -> ~ In x s'.
- Parameter diff_3 In x s -> ~ In x s' -> In x (diff s s').
+ Parameter diff_1 : In x (diff s s') -> In x s.
+ Parameter diff_2 : In x (diff s s') -> ~ In x s'.
+ Parameter diff_3 : In x s -> ~ In x s' -> In x (diff s s').
(** Specification of [fold] *)
- Parameter fold_1 forall (A : Set) (i : A) (f : elt -> A -> A),
+ Parameter fold_1 : forall (A : Set) (i : A) (f : elt -> A -> A),
fold f s i = fold_left (fun a e => f e a) (elements s) i.
(** Specification of [cardinal] *)
- Parameter cardinal_1 cardinal s = length (elements s).
+ Parameter cardinal_1 : cardinal s = length (elements s).
Section Filter.
- Variable f elt -> bool.
+ Variable f : elt -> bool.
(** Specification of [filter] *)
- Parameter filter_1 compat_bool E.eq f -> In x (filter f s) -> In x s.
- Parameter filter_2 compat_bool E.eq f -> In x (filter f s) -> f x = true.
- Parameter filter_3
+ Parameter filter_1 : compat_bool E.eq f -> In x (filter f s) -> In x s.
+ Parameter filter_2 : compat_bool E.eq f -> In x (filter f s) -> f x = true.
+ Parameter filter_3 :
compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s).
(** Specification of [for_all] *)
- Parameter for_all_1
+ Parameter for_all_1 :
compat_bool E.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
- Parameter for_all_2
+ Parameter for_all_2 :
compat_bool E.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.
(** Specification of [exists] *)
- Parameter exists_1
+ Parameter exists_1 :
compat_bool E.eq f ->
Exists (fun x => f x = true) s -> exists_ f s = true.
- Parameter exists_2
+ Parameter exists_2 :
compat_bool E.eq f ->
exists_ f s = true -> Exists (fun x => f x = true) s.
(** Specification of [partition] *)
- Parameter partition_1
+ Parameter partition_1 :
compat_bool E.eq f -> Equal (fst (partition f s)) (filter f s).
- Parameter partition_2
+ Parameter partition_2 :
compat_bool E.eq f ->
Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
(** Specification of [elements] *)
- Parameter elements_1 In x s -> InA E.eq x (elements s).
- Parameter elements_2 InA E.eq x (elements s) -> In x s.
+ Parameter elements_1 : In x s -> InA E.eq x (elements s).
+ Parameter elements_2 : InA E.eq x (elements s) -> In x s.
(** Specification of [choose] *)
- Parameter choose_1 choose s = Some x -> In x s.
- Parameter choose_2 choose s = None -> Empty s.
+ Parameter choose_1 : choose s = Some x -> In x s.
+ Parameter choose_2 : choose s = None -> Empty s.
End Filter.
End Spec.
diff --git a/theories/FSets/FSetWeakList.v b/theories/FSets/FSetWeakList.v
index 0a13ee78e..bcd966f9a 100644
--- a/theories/FSets/FSetWeakList.v
+++ b/theories/FSets/FSetWeakList.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: FSetWeakList.v,v 1.8 2006/03/09 18:34:51 letouzey Exp $ *)
(** * Finite sets library *)
@@ -23,116 +23,116 @@ Unset Strict Implicit.
The specs are proved under the additional condition of no redundancy.
And the functions returning sets are proved to preserve this invariant. *)
-Module Raw (X DecidableType).
+Module Raw (X: DecidableType).
- Module E = X.
+ Module E := X.
- Definition elt = X.t.
- Definition t = list elt.
+ Definition elt := X.t.
+ Definition t := list elt.
- Definition empty t := nil.
+ Definition empty : t := nil.
- Definition is_empty (l t) : bool := if l then true else false.
+ Definition is_empty (l : t) : bool := if l then true else false.
(** ** The set operations. *)
- Fixpoint mem (x elt) (s : t) {struct s} : bool :=
+ Fixpoint mem (x : elt) (s : t) {struct s} : bool :=
match s with
| nil => false
- | y : l =>
+ | y :: l =>
if X.eq_dec x y then true else mem x l
end.
- Fixpoint add (x elt) (s : t) {struct s} : t :=
+ Fixpoint add (x : elt) (s : t) {struct s} : t :=
match s with
- | nil => x : nil
- | y : l =>
- if X.eq_dec x y then s else y : add x l
+ | nil => x :: nil
+ | y :: l =>
+ if X.eq_dec x y then s else y :: add x l
end.
- Definition singleton (x elt) : t := x :: nil.
+ Definition singleton (x : elt) : t := x :: nil.
- Fixpoint remove (x elt) (s : t) {struct s} : t :=
+ Fixpoint remove (x : elt) (s : t) {struct s} : t :=
match s with
| nil => nil
- | y : l =>
- if X.eq_dec x y then l else y : remove x l
+ | y :: l =>
+ if X.eq_dec x y then l else y :: remove x l
end.
- Fixpoint fold (B Set) (f : elt -> B -> B) (s : t) {struct s} :
- B -> B = fun i => match s with
+ Fixpoint fold (B : Set) (f : elt -> B -> B) (s : t) {struct s} :
+ B -> B := fun i => match s with
| nil => i
- | x : l => fold f l (f x i)
+ | x :: l => fold f l (f x i)
end.
- Definition union (s t) : t -> t := fold add s.
+ Definition union (s : t) : t -> t := fold add s.
- Definition diff (s s' t) : t := fold remove s' s.
+ Definition diff (s s' : t) : t := fold remove s' s.
- Definition inter (s s' t) : t :=
+ Definition inter (s s': t) : t :=
fold (fun x s => if mem x s' then add x s else s) s nil.
- Definition subset (s s' t) : bool := is_empty (diff s s').
+ Definition subset (s s' : t) : bool := is_empty (diff s s').
- Definition equal (s s' t) : bool := andb (subset s s') (subset s' s).
+ Definition equal (s s' : t) : bool := andb (subset s s') (subset s' s).
- Fixpoint filter (f elt -> bool) (s : t) {struct s} : t :=
+ Fixpoint filter (f : elt -> bool) (s : t) {struct s} : t :=
match s with
| nil => nil
- | x : l => if f x then x :: filter f l else filter f l
+ | x :: l => if f x then x :: filter f l else filter f l
end.
- Fixpoint for_all (f elt -> bool) (s : t) {struct s} : bool :=
+ Fixpoint for_all (f : elt -> bool) (s : t) {struct s} : bool :=
match s with
| nil => true
- | x : l => if f x then for_all f l else false
+ | x :: l => if f x then for_all f l else false
end.
- Fixpoint exists_ (f elt -> bool) (s : t) {struct s} : bool :=
+ Fixpoint exists_ (f : elt -> bool) (s : t) {struct s} : bool :=
match s with
| nil => false
- | x : l => if f x then true else exists_ f l
+ | x :: l => if f x then true else exists_ f l
end.
- Fixpoint partition (f elt -> bool) (s : t) {struct s} :
- t * t =
+ Fixpoint partition (f : elt -> bool) (s : t) {struct s} :
+ t * t :=
match s with
| nil => (nil, nil)
- | x : l =>
- let (s1, s2) = partition f l in
- if f x then (x : s1, s2) else (s1, x :: s2)
+ | x :: l =>
+ let (s1, s2) := partition f l in
+ if f x then (x :: s1, s2) else (s1, x :: s2)
end.
- Definition cardinal (s t) : nat := length s.
+ Definition cardinal (s : t) : nat := length s.
- Definition elements (s t) : list elt := s.
+ Definition elements (s : t) : list elt := s.
- Definition choose (s t) : option elt :=
+ Definition choose (s : t) : option elt :=
match s with
| nil => None
- | x:_ => Some x
+ | x::_ => Some x
end.
(** ** Proofs of set operation specifications. *)
- Notation NoRedun = (noredunA X.eq).
- Notation In = (InA X.eq).
+ Notation NoRedun := (noredunA X.eq).
+ Notation In := (InA X.eq).
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) s := forall x, In x s -> P x.
- Definition Exists (P elt -> Prop) s := exists x, In x s /\ P x.
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
+ Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.
- Lemma In_eq
- forall (s t) (x y : elt), X.eq x y -> In x s -> In y s.
+ Lemma In_eq :
+ forall (s : t) (x y : elt), X.eq x y -> In x s -> In y s.
Proof.
intros s x y; do 2 setoid_rewrite InA_alt; firstorder eauto.
Qed.
Hint Immediate In_eq.
- Lemma mem_1
- forall (s t)(x : elt), In x s -> mem x s = true.
+ Lemma mem_1 :
+ forall (s : t)(x : elt), In x s -> mem x s = true.
Proof.
induction s; intros.
inversion H.
@@ -140,7 +140,7 @@ Module Raw (X DecidableType).
inversion_clear H; auto.
Qed.
- Lemma mem_2 forall (s : t) (x : elt), mem x s = true -> In x s.
+ Lemma mem_2 : forall (s : t) (x : elt), mem x s = true -> In x s.
Proof.
induction s.
intros; inversion H.
@@ -148,8 +148,8 @@ Module Raw (X DecidableType).
destruct (X.eq_dec x a); firstorder; discriminate.
Qed.
- Lemma add_1
- forall (s t) (Hs : NoRedun s) (x y : elt), X.eq x y -> In y (add x s).
+ Lemma add_1 :
+ forall (s : t) (Hs : NoRedun s) (x y : elt), X.eq x y -> In y (add x s).
Proof.
induction s.
simpl; intuition.
@@ -158,8 +158,8 @@ Module Raw (X DecidableType).
eauto.
Qed.
- Lemma add_2
- forall (s t) (Hs : NoRedun s) (x y : elt), In y s -> In y (add x s).
+ Lemma add_2 :
+ forall (s : t) (Hs : NoRedun s) (x y : elt), In y s -> In y (add x s).
Proof.
induction s.
simpl; intuition.
@@ -167,8 +167,8 @@ Module Raw (X DecidableType).
inversion_clear Hs; eauto; inversion_clear H; intuition.
Qed.
- Lemma add_3
- forall (s t) (Hs : NoRedun s) (x y : elt),
+ Lemma add_3 :
+ forall (s : t) (Hs : NoRedun s) (x y : elt),
~ X.eq x y -> In y (add x s) -> In y s.
Proof.
induction s.
@@ -179,8 +179,8 @@ Module Raw (X DecidableType).
absurd (X.eq x y); auto.
Qed.
- Lemma add_unique
- forall (s t) (Hs : NoRedun s)(x:elt), NoRedun (add x s).
+ Lemma add_unique :
+ forall (s : t) (Hs : NoRedun s)(x:elt), NoRedun (add x s).
Proof.
induction s.
simpl; intuition.
@@ -196,8 +196,8 @@ Module Raw (X DecidableType).
eapply add_3; eauto.
Qed.
- Lemma remove_1
- forall (s t) (Hs : NoRedun s) (x y : elt), X.eq x y -> ~ In y (remove x s).
+ Lemma remove_1 :
+ forall (s : t) (Hs : NoRedun s) (x y : elt), X.eq x y -> ~ In y (remove x s).
Proof.
simple induction s.
simpl; red; intros; inversion H0.
@@ -207,8 +207,8 @@ Module Raw (X DecidableType).
inversion_clear H1; eauto.
Qed.
- Lemma remove_2
- forall (s t) (Hs : NoRedun s) (x y : elt),
+ Lemma remove_2 :
+ forall (s : t) (Hs : NoRedun s) (x y : elt),
~ X.eq x y -> In y s -> In y (remove x s).
Proof.
simple induction s.
@@ -218,8 +218,8 @@ Module Raw (X DecidableType).
absurd (X.eq x y); eauto.
Qed.
- Lemma remove_3
- forall (s t) (Hs : NoRedun s) (x y : elt), In y (remove x s) -> In y s.
+ Lemma remove_3 :
+ forall (s : t) (Hs : NoRedun s) (x y : elt), In y (remove x s) -> In y s.
Proof.
simple induction s.
simpl; intuition.
@@ -227,8 +227,8 @@ Module Raw (X DecidableType).
inversion_clear Hs; inversion_clear H; firstorder.
Qed.
- Lemma remove_unique
- forall (s t) (Hs : NoRedun s) (x : elt), NoRedun (remove x s).
+ Lemma remove_unique :
+ forall (s : t) (Hs : NoRedun s) (x : elt), NoRedun (remove x s).
Proof.
simple induction s.
simpl; intuition.
@@ -239,69 +239,69 @@ Module Raw (X DecidableType).
eapply remove_3; eauto.
Qed.
- Lemma singleton_unique forall x : elt, NoRedun (singleton x).
+ Lemma singleton_unique : forall x : elt, NoRedun (singleton x).
Proof.
unfold singleton; simpl; constructor; auto; intro H; inversion H.
Qed.
- Lemma singleton_1 forall x y : elt, In y (singleton x) -> X.eq x y.
+ Lemma singleton_1 : forall x y : elt, In y (singleton x) -> X.eq x y.
Proof.
unfold singleton; simpl; intuition.
inversion_clear H; auto; inversion H0.
Qed.
- Lemma singleton_2 forall x y : elt, X.eq x y -> In y (singleton x).
+ Lemma singleton_2 : forall x y : elt, X.eq x y -> In y (singleton x).
Proof.
unfold singleton; simpl; intuition.
Qed.
- Lemma empty_unique NoRedun empty.
+ Lemma empty_unique : NoRedun empty.
Proof.
unfold empty; constructor.
Qed.
- Lemma empty_1 Empty empty.
+ Lemma empty_1 : Empty empty.
Proof.
unfold Empty, empty; intuition; inversion H.
Qed.
- Lemma is_empty_1 forall s : t, Empty s -> is_empty s = true.
+ Lemma is_empty_1 : forall s : t, Empty s -> is_empty s = true.
Proof.
unfold Empty; intro s; case s; simpl; intuition.
elim (H e); auto.
Qed.
- Lemma is_empty_2 forall s : t, is_empty s = true -> Empty s.
+ Lemma is_empty_2 : forall s : t, is_empty s = true -> Empty s.
Proof.
unfold Empty; intro s; case s; simpl; intuition;
inversion H0.
Qed.
- Lemma elements_1 forall (s : t) (x : elt), In x s -> In x (elements s).
+ Lemma elements_1 : forall (s : t) (x : elt), In x s -> In x (elements s).
Proof.
unfold elements; auto.
Qed.
- Lemma elements_2 forall (s : t) (x : elt), In x (elements s) -> In x s.
+ Lemma elements_2 : forall (s : t) (x : elt), In x (elements s) -> In x s.
Proof.
unfold elements; auto.
Qed.
- Lemma elements_3 forall (s : t) (Hs : NoRedun s), NoRedun (elements s).
+ Lemma elements_3 : forall (s : t) (Hs : NoRedun s), NoRedun (elements s).
Proof.
unfold elements; auto.
Qed.
- Lemma fold_1
- forall (s t) (Hs : NoRedun s) (A : Set) (i : A) (f : elt -> A -> A),
+ Lemma fold_1 :
+ forall (s : t) (Hs : NoRedun s) (A : Set) (i : A) (f : elt -> A -> A),
fold f s i = fold_left (fun a e => f e a) (elements s) i.
Proof.
induction s; simpl; auto; intros.
inversion_clear Hs; auto.
Qed.
- Lemma union_unique
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s'), NoRedun (union s s').
+ Lemma union_unique :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s'), NoRedun (union s s').
Proof.
unfold union; induction s; simpl; auto; intros.
inversion_clear Hs.
@@ -309,8 +309,8 @@ Module Raw (X DecidableType).
apply add_unique; auto.
Qed.
- Lemma union_1
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma union_1 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (union s s') -> In x s \/ In x s'.
Proof.
unfold union; induction s; simpl; auto; intros.
@@ -321,8 +321,8 @@ Module Raw (X DecidableType).
right; eapply add_3; eauto.
Qed.
- Lemma union_0
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma union_0 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x s \/ In x s' -> In x (union s s').
Proof.
unfold union; induction s; simpl; auto; intros.
@@ -337,25 +337,25 @@ Module Raw (X DecidableType).
right; apply add_2; auto.
Qed.
- Lemma union_2
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma union_2 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x s -> In x (union s s').
Proof.
intros; apply union_0; auto.
Qed.
- Lemma union_3
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma union_3 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x s' -> In x (union s s').
Proof.
intros; apply union_0; auto.
Qed.
- Lemma inter_unique
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s'), NoRedun (inter s s').
+ Lemma inter_unique :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s'), NoRedun (inter s s').
Proof.
unfold inter; intros s.
- set (acc = nil (A:=elt)).
+ set (acc := nil (A:=elt)).
assert (NoRedun acc) by (unfold acc; auto).
clearbody acc; generalize H; clear H; generalize acc; clear acc.
induction s; simpl; auto; intros.
@@ -365,12 +365,12 @@ Module Raw (X DecidableType).
apply add_unique; auto.
Qed.
- Lemma inter_0
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma inter_0 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (inter s s') -> In x s /\ In x s'.
Proof.
unfold inter; intros.
- set (acc = nil (A:=elt)) in *.
+ set (acc := nil (A:=elt)) in *.
assert (NoRedun acc) by (unfold acc; auto).
cut ((In x s /\ In x s') \/ In x acc).
destruct 1; auto.
@@ -392,29 +392,29 @@ Module Raw (X DecidableType).
left; intuition.
Qed.
- Lemma inter_1
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma inter_1 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (inter s s') -> In x s.
Proof.
intros; cut (In x s /\ In x s'); [ intuition | apply inter_0; auto ].
Qed.
- Lemma inter_2
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma inter_2 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (inter s s') -> In x s'.
Proof.
intros; cut (In x s /\ In x s'); [ intuition | apply inter_0; auto ].
Qed.
- Lemma inter_3
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma inter_3 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x s -> In x s' -> In x (inter s s').
Proof.
intros s s' Hs Hs' x.
- cut (((In x s /\ In x s')\/ In x (nil (A=elt))) -> In x (inter s s')).
+ cut (((In x s /\ In x s')\/ In x (nil (A:=elt))) -> In x (inter s s')).
intuition.
unfold inter.
- set (acc = nil (A:=elt)) in *.
+ set (acc := nil (A:=elt)) in *.
assert (NoRedun acc) by (unfold acc; auto).
clearbody acc.
generalize H Hs' Hs; clear H Hs Hs'.
@@ -440,8 +440,8 @@ Module Raw (X DecidableType).
discriminate.
Qed.
- Lemma diff_unique
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s'), NoRedun (diff s s').
+ Lemma diff_unique :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s'), NoRedun (diff s s').
Proof.
unfold diff; intros s s' Hs; generalize s Hs; clear Hs s.
induction s'; simpl; auto; intros.
@@ -450,8 +450,8 @@ Module Raw (X DecidableType).
apply remove_unique; auto.
Qed.
- Lemma diff_0
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma diff_0 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (diff s s') -> In x s /\ ~ In x s'.
Proof.
unfold diff; intros s s' Hs; generalize s Hs; clear Hs s.
@@ -467,22 +467,22 @@ Module Raw (X DecidableType).
destruct (remove_1 Hs (X.eq_sym H5) H2).
Qed.
- Lemma diff_1
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma diff_1 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (diff s s') -> In x s.
Proof.
intros; cut (In x s /\ ~ In x s'); [ intuition | apply diff_0; auto].
Qed.
- Lemma diff_2
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma diff_2 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x (diff s s') -> ~ In x s'.
Proof.
intros; cut (In x s /\ ~ In x s'); [ intuition | apply diff_0; auto].
Qed.
- Lemma diff_3
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
+ Lemma diff_3 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s') (x : elt),
In x s -> ~ In x s' -> In x (diff s s').
Proof.
unfold diff; intros s s' Hs; generalize s Hs; clear Hs s.
@@ -493,8 +493,8 @@ Module Raw (X DecidableType).
apply remove_2; auto.
Qed.
- Lemma subset_1
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s'),
+ Lemma subset_1 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s'),
Subset s s' -> subset s s' = true.
Proof.
unfold subset, Subset; intros.
@@ -506,7 +506,7 @@ Module Raw (X DecidableType).
eapply diff_1; eauto.
Qed.
- Lemma subset_2 forall (s s' : t)(Hs : NoRedun s) (Hs' : NoRedun s'),
+ Lemma subset_2 : forall (s s' : t)(Hs : NoRedun s) (Hs' : NoRedun s'),
subset s s' = true -> Subset s s'.
Proof.
unfold subset, Subset; intros.
@@ -518,15 +518,15 @@ Module Raw (X DecidableType).
apply diff_3; intuition.
Qed.
- Lemma equal_1
- forall (s s' t) (Hs : NoRedun s) (Hs' : NoRedun s'),
+ Lemma equal_1 :
+ forall (s s' : t) (Hs : NoRedun s) (Hs' : NoRedun s'),
Equal s s' -> equal s s' = true.
Proof.
unfold Equal, equal; intros.
apply andb_true_intro; split; apply subset_1; firstorder.
Qed.
- Lemma equal_2 forall (s s' : t)(Hs : NoRedun s) (Hs' : NoRedun s'),
+ Lemma equal_2 : forall (s s' : t)(Hs : NoRedun s) (Hs' : NoRedun s'),
equal s s' = true -> Equal s s'.
Proof.
unfold Equal, equal; intros.
@@ -534,27 +534,27 @@ Module Raw (X DecidableType).
split; apply subset_2; auto.
Qed.
- Definition choose_1
- forall (s t) (x : elt), choose s = Some x -> In x s.
+ Definition choose_1 :
+ forall (s : t) (x : elt), choose s = Some x -> In x s.
Proof.
destruct s; simpl; intros; inversion H; auto.
Qed.
- Definition choose_2 forall s : t, choose s = None -> Empty s.
+ Definition choose_2 : forall s : t, choose s = None -> Empty s.
Proof.
destruct s; simpl; intros.
intros x H0; inversion H0.
inversion H.
Qed.
- Lemma cardinal_1
- forall (s t) (Hs : NoRedun s), cardinal s = length (elements s).
+ Lemma cardinal_1 :
+ forall (s : t) (Hs : NoRedun s), cardinal s = length (elements s).
Proof.
auto.
Qed.
- Lemma filter_1
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_1 :
+ forall (s : t) (x : elt) (f : elt -> bool),
In x (filter f s) -> In x s.
Proof.
simple induction s; simpl.
@@ -567,8 +567,8 @@ Module Raw (X DecidableType).
constructor 2; apply (Hrec a f); trivial.
Qed.
- Lemma filter_2
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_2 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool X.eq f -> In x (filter f s) -> f x = true.
Proof.
simple induction s; simpl.
@@ -579,8 +579,8 @@ Module Raw (X DecidableType).
symmetry; auto.
Qed.
- Lemma filter_3
- forall (s t) (x : elt) (f : elt -> bool),
+ Lemma filter_3 :
+ forall (s : t) (x : elt) (f : elt -> bool),
compat_bool X.eq f -> In x s -> f x = true -> In x (filter f s).
Proof.
simple induction s; simpl.
@@ -592,8 +592,8 @@ Module Raw (X DecidableType).
rewrite <- (H a (X.eq_sym H1)); intros; discriminate.
Qed.
- Lemma filter_unique
- forall (s t) (Hs : NoRedun s) (f : elt -> bool), NoRedun (filter f s).
+ Lemma filter_unique :
+ forall (s : t) (Hs : NoRedun s) (f : elt -> bool), NoRedun (filter f s).
Proof.
simple induction s; simpl.
auto.
@@ -605,8 +605,8 @@ Module Raw (X DecidableType).
Qed.
- Lemma for_all_1
- forall (s t) (f : elt -> bool),
+ Lemma for_all_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
Proof.
@@ -617,8 +617,8 @@ Module Raw (X DecidableType).
intros; rewrite (H x); auto.
Qed.
- Lemma for_all_2
- forall (s t) (f : elt -> bool),
+ Lemma for_all_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.
Proof.
@@ -633,8 +633,8 @@ Module Raw (X DecidableType).
rewrite (Hf a x); auto.
Qed.
- Lemma exists_1
- forall (s t) (f : elt -> bool),
+ Lemma exists_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f -> Exists (fun x => f x = true) s -> exists_ f s = true.
Proof.
simple induction s; simpl; auto; unfold Exists.
@@ -651,8 +651,8 @@ Module Raw (X DecidableType).
exists a; auto.
Qed.
- Lemma exists_2
- forall (s t) (f : elt -> bool),
+ Lemma exists_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f -> exists_ f s = true -> Exists (fun x => f x = true) s.
Proof.
simple induction s; simpl; auto; unfold Exists.
@@ -664,8 +664,8 @@ Module Raw (X DecidableType).
exists a; auto.
Qed.
- Lemma partition_1
- forall (s t) (f : elt -> bool),
+ Lemma partition_1 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f -> Equal (fst (partition f s)) (filter f s).
Proof.
simple induction s; simpl; auto; unfold Equal.
@@ -676,8 +676,8 @@ Module Raw (X DecidableType).
case (f x); simpl; firstorder; inversion H0; intros; firstorder.
Qed.
- Lemma partition_2
- forall (s t) (f : elt -> bool),
+ Lemma partition_2 :
+ forall (s : t) (f : elt -> bool),
compat_bool X.eq f ->
Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
Proof.
@@ -689,8 +689,8 @@ Module Raw (X DecidableType).
case (f x); simpl; firstorder; inversion H0; intros; firstorder.
Qed.
- Lemma partition_aux_1
- forall (s t) (Hs : NoRedun s) (f : elt -> bool)(x:elt),
+ Lemma partition_aux_1 :
+ forall (s : t) (Hs : NoRedun s) (f : elt -> bool)(x:elt),
In x (fst (partition f s)) -> In x s.
Proof.
induction s; simpl; auto; intros.
@@ -700,8 +700,8 @@ Module Raw (X DecidableType).
inversion_clear H; auto.
Qed.
- Lemma partition_aux_2
- forall (s t) (Hs : NoRedun s) (f : elt -> bool)(x:elt),
+ Lemma partition_aux_2 :
+ forall (s : t) (Hs : NoRedun s) (f : elt -> bool)(x:elt),
In x (snd (partition f s)) -> In x s.
Proof.
induction s; simpl; auto; intros.
@@ -711,8 +711,8 @@ Module Raw (X DecidableType).
inversion_clear H; auto.
Qed.
- Lemma partition_unique_1
- forall (s t) (Hs : NoRedun s) (f : elt -> bool), NoRedun (fst (partition f s)).
+ Lemma partition_unique_1 :
+ forall (s : t) (Hs : NoRedun s) (f : elt -> bool), NoRedun (fst (partition f s)).
Proof.
simple induction s; simpl.
auto.
@@ -722,8 +722,8 @@ Module Raw (X DecidableType).
case (f x); case (partition f l); simpl; auto.
Qed.
- Lemma partition_unique_2
- forall (s t) (Hs : NoRedun s) (f : elt -> bool), NoRedun (snd (partition f s)).
+ Lemma partition_unique_2 :
+ forall (s : t) (Hs : NoRedun s) (f : elt -> bool), NoRedun (snd (partition f s)).
Proof.
simple induction s; simpl.
auto.
@@ -733,19 +733,19 @@ Module Raw (X DecidableType).
case (f x); case (partition f l); simpl; auto.
Qed.
- Definition eq t -> t -> Prop := Equal.
+ Definition eq : t -> t -> Prop := Equal.
- Lemma eq_refl forall s : t, eq s s.
+ Lemma eq_refl : forall s : t, eq s s.
Proof.
unfold eq, Equal; intuition.
Qed.
- Lemma eq_sym forall s s' : t, eq s s' -> eq s' s.
+ Lemma eq_sym : forall s s' : t, eq s s' -> eq s' s.
Proof.
unfold eq, Equal; firstorder.
Qed.
- Lemma eq_trans forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
+ Lemma eq_trans : forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
Proof.
unfold eq, Equal; firstorder.
Qed.
@@ -757,117 +757,117 @@ End Raw.
Now, in order to really provide a functor implementing [S], we
need to encapsulate everything into a type of lists without redundancy. *)
-Module Make (X DecidableType) <: S with Module E := X.
+Module Make (X: DecidableType) <: S with Module E := X.
- Module E = X.
- Module Raw = Raw X.
+ Module E := X.
+ Module Raw := Raw X.
- Record slist Set := {this :> Raw.t; unique : noredunA X.eq this}.
- Definition t = slist.
- Definition elt = X.t.
+ Record slist : Set := {this :> Raw.t; unique : noredunA X.eq this}.
+ Definition t := slist.
+ Definition elt := X.t.
- Definition In (x elt) (s : t) := InA X.eq x s.(this).
- Definition Equal s s' = forall a : elt, In a s <-> In a s'.
- Definition Subset s s' = forall a : elt, In a s -> In a s'.
- Definition Empty s = forall a : elt, ~ In a s.
- Definition For_all (P elt -> Prop) (s : t) :=
- forall x elt, In x s -> P x.
- Definition Exists (P elt -> Prop) (s : t) := exists x : elt, In x s /\ P x.
+ Definition In (x : elt) (s : t) := InA X.eq x s.(this).
+ Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+ Definition Subset s s' := forall a : elt, In a s -> In a s'.
+ Definition Empty s := forall a : elt, ~ In a s.
+ Definition For_all (P : elt -> Prop) (s : t) :=
+ forall x : elt, In x s -> P x.
+ Definition Exists (P : elt -> Prop) (s : t) := exists x : elt, In x s /\ P x.
- Definition In_1 (s t) := Raw.In_eq (s:=s).
+ Definition In_1 (s : t) := Raw.In_eq (s:=s).
- Definition mem (x elt) (s : t) := Raw.mem x s.
- Definition mem_1 (s t) := Raw.mem_1 (s:=s).
- Definition mem_2 (s t) := Raw.mem_2 (s:=s).
-
- Definition add x s = Build_slist (Raw.add_unique (unique s) x).
- Definition add_1 (s t) := Raw.add_1 (unique s).
- Definition add_2 (s t) := Raw.add_2 (unique s).
- Definition add_3 (s t) := Raw.add_3 (unique s).
-
- Definition remove x s = Build_slist (Raw.remove_unique (unique s) x).
- Definition remove_1 (s t) := Raw.remove_1 (unique s).
- Definition remove_2 (s t) := Raw.remove_2 (unique s).
- Definition remove_3 (s t) := Raw.remove_3 (unique s).
+ Definition mem (x : elt) (s : t) := Raw.mem x s.
+ Definition mem_1 (s : t) := Raw.mem_1 (s:=s).
+ Definition mem_2 (s : t) := Raw.mem_2 (s:=s).
+
+ Definition add x s := Build_slist (Raw.add_unique (unique s) x).
+ Definition add_1 (s : t) := Raw.add_1 (unique s).
+ Definition add_2 (s : t) := Raw.add_2 (unique s).
+ Definition add_3 (s : t) := Raw.add_3 (unique s).
+
+ Definition remove x s := Build_slist (Raw.remove_unique (unique s) x).
+ Definition remove_1 (s : t) := Raw.remove_1 (unique s).
+ Definition remove_2 (s : t) := Raw.remove_2 (unique s).
+ Definition remove_3 (s : t) := Raw.remove_3 (unique s).
- Definition singleton x = Build_slist (Raw.singleton_unique x).
- Definition singleton_1 = Raw.singleton_1.
- Definition singleton_2 = Raw.singleton_2.
+ Definition singleton x := Build_slist (Raw.singleton_unique x).
+ Definition singleton_1 := Raw.singleton_1.
+ Definition singleton_2 := Raw.singleton_2.
- Definition union (s s' t) :=
+ Definition union (s s' : t) :=
Build_slist (Raw.union_unique (unique s) (unique s')).
- Definition union_1 (s s' t) := Raw.union_1 (unique s) (unique s').
- Definition union_2 (s s' t) := Raw.union_2 (unique s) (unique s').
- Definition union_3 (s s' t) := Raw.union_3 (unique s) (unique s').
+ Definition union_1 (s s' : t) := Raw.union_1 (unique s) (unique s').
+ Definition union_2 (s s' : t) := Raw.union_2 (unique s) (unique s').
+ Definition union_3 (s s' : t) := Raw.union_3 (unique s) (unique s').
- Definition inter (s s' t) :=
+ Definition inter (s s' : t) :=
Build_slist (Raw.inter_unique (unique s) (unique s')).
- Definition inter_1 (s s' t) := Raw.inter_1 (unique s) (unique s').
- Definition inter_2 (s s' t) := Raw.inter_2 (unique s) (unique s').
- Definition inter_3 (s s' t) := Raw.inter_3 (unique s) (unique s').
+ Definition inter_1 (s s' : t) := Raw.inter_1 (unique s) (unique s').
+ Definition inter_2 (s s' : t) := Raw.inter_2 (unique s) (unique s').
+ Definition inter_3 (s s' : t) := Raw.inter_3 (unique s) (unique s').
- Definition diff (s s' t) :=
+ Definition diff (s s' : t) :=
Build_slist (Raw.diff_unique (unique s) (unique s')).
- Definition diff_1 (s s' t) := Raw.diff_1 (unique s) (unique s').
- Definition diff_2 (s s' t) := Raw.diff_2 (unique s) (unique s').
- Definition diff_3 (s s' t) := Raw.diff_3 (unique s) (unique s').
+ Definition diff_1 (s s' : t) := Raw.diff_1 (unique s) (unique s').
+ Definition diff_2 (s s' : t) := Raw.diff_2 (unique s) (unique s').
+ Definition diff_3 (s s' : t) := Raw.diff_3 (unique s) (unique s').
- Definition equal (s s' t) := Raw.equal s s'.
- Definition equal_1 (s s' t) := Raw.equal_1 (unique s) (unique s').
- Definition equal_2 (s s' t) := Raw.equal_2 (unique s) (unique s').
+ Definition equal (s s' : t) := Raw.equal s s'.
+ Definition equal_1 (s s' : t) := Raw.equal_1 (unique s) (unique s').
+ Definition equal_2 (s s' : t) := Raw.equal_2 (unique s) (unique s').
- Definition subset (s s' t) := Raw.subset s s'.
- Definition subset_1 (s s' t) := Raw.subset_1 (unique s) (unique s').
- Definition subset_2 (s s' t) := Raw.subset_2 (unique s) (unique s').
+ Definition subset (s s' : t) := Raw.subset s s'.
+ Definition subset_1 (s s' : t) := Raw.subset_1 (unique s) (unique s').
+ Definition subset_2 (s s' : t) := Raw.subset_2 (unique s) (unique s').
- Definition empty = Build_slist Raw.empty_unique.
- Definition empty_1 = Raw.empty_1.
+ Definition empty := Build_slist Raw.empty_unique.
+ Definition empty_1 := Raw.empty_1.
- Definition is_empty (s t) := Raw.is_empty s.
- Definition is_empty_1 (s t) := Raw.is_empty_1 (s:=s).
- Definition is_empty_2 (s t) := Raw.is_empty_2 (s:=s).
-
- Definition elements (s t) := Raw.elements s.
- Definition elements_1 (s t) := Raw.elements_1 (s:=s).
- Definition elements_2 (s t) := Raw.elements_2 (s:=s).
- Definition elements_3 (s t) := Raw.elements_3 (unique s).
+ Definition is_empty (s : t) := Raw.is_empty s.
+ Definition is_empty_1 (s : t) := Raw.is_empty_1 (s:=s).
+ Definition is_empty_2 (s : t) := Raw.is_empty_2 (s:=s).
+
+ Definition elements (s : t) := Raw.elements s.
+ Definition elements_1 (s : t) := Raw.elements_1 (s:=s).
+ Definition elements_2 (s : t) := Raw.elements_2 (s:=s).
+ Definition elements_3 (s : t) := Raw.elements_3 (unique s).
- Definition choose (st) := Raw.choose s.
- Definition choose_1 (s t) := Raw.choose_1 (s:=s).
- Definition choose_2 (s t) := Raw.choose_2 (s:=s).
+ Definition choose (s:t) := Raw.choose s.
+ Definition choose_1 (s : t) := Raw.choose_1 (s:=s).
+ Definition choose_2 (s : t) := Raw.choose_2 (s:=s).
- Definition fold (B Set) (f : elt -> B -> B) (s : t) := Raw.fold (B:=B) f s.
- Definition fold_1 (s t) := Raw.fold_1 (unique s).
+ Definition fold (B : Set) (f : elt -> B -> B) (s : t) := Raw.fold (B:=B) f s.
+ Definition fold_1 (s : t) := Raw.fold_1 (unique s).
- Definition cardinal (s t) := Raw.cardinal s.
- Definition cardinal_1 (s t) := Raw.cardinal_1 (unique s).
+ Definition cardinal (s : t) := Raw.cardinal s.
+ Definition cardinal_1 (s : t) := Raw.cardinal_1 (unique s).
- Definition filter (f elt -> bool) (s : t) :=
+ Definition filter (f : elt -> bool) (s : t) :=
Build_slist (Raw.filter_unique (unique s) f).
- Definition filter_1 (s t)(x:elt)(f: elt -> bool)(H:compat_bool X.eq f) :=
+ Definition filter_1 (s : t)(x:elt)(f: elt -> bool)(H:compat_bool X.eq f) :=
@Raw.filter_1 s x f.
- Definition filter_2 (s t) := Raw.filter_2 (s:=s).
- Definition filter_3 (s t) := Raw.filter_3 (s:=s).
+ Definition filter_2 (s : t) := Raw.filter_2 (s:=s).
+ Definition filter_3 (s : t) := Raw.filter_3 (s:=s).
- Definition for_all (f elt -> bool) (s : t) := Raw.for_all f s.
- Definition for_all_1 (s t) := Raw.for_all_1 (s:=s).
- Definition for_all_2 (s t) := Raw.for_all_2 (s:=s).
-
- Definition exists_ (f elt -> bool) (s : t) := Raw.exists_ f s.
- Definition exists_1 (s t) := Raw.exists_1 (s:=s).
- Definition exists_2 (s t) := Raw.exists_2 (s:=s).
-
- Definition partition (f elt -> bool) (s : t) :=
- let p = Raw.partition f s in
- (Build_slist (this=fst p) (Raw.partition_unique_1 (unique s) f),
- Build_slist (this=snd p) (Raw.partition_unique_2 (unique s) f)).
- Definition partition_1 (s t) := Raw.partition_1 s.
- Definition partition_2 (s t) := Raw.partition_2 s.
-
- Definition eq (s s' t) := Raw.eq s s'.
- Definition eq_refl (s t) := Raw.eq_refl s.
- Definition eq_sym (s s' t) := Raw.eq_sym (s:=s) (s':=s').
- Definition eq_trans (s s' s'' t) :=
- Raw.eq_trans (s=s) (s':=s') (s'':=s'').
+ Definition for_all (f : elt -> bool) (s : t) := Raw.for_all f s.
+ Definition for_all_1 (s : t) := Raw.for_all_1 (s:=s).
+ Definition for_all_2 (s : t) := Raw.for_all_2 (s:=s).
+
+ Definition exists_ (f : elt -> bool) (s : t) := Raw.exists_ f s.
+ Definition exists_1 (s : t) := Raw.exists_1 (s:=s).
+ Definition exists_2 (s : t) := Raw.exists_2 (s:=s).
+
+ Definition partition (f : elt -> bool) (s : t) :=
+ let p := Raw.partition f s in
+ (Build_slist (this:=fst p) (Raw.partition_unique_1 (unique s) f),
+ Build_slist (this:=snd p) (Raw.partition_unique_2 (unique s) f)).
+ Definition partition_1 (s : t) := Raw.partition_1 s.
+ Definition partition_2 (s : t) := Raw.partition_2 s.
+
+ Definition eq (s s' : t) := Raw.eq s s'.
+ Definition eq_refl (s : t) := Raw.eq_refl s.
+ Definition eq_sym (s s' : t) := Raw.eq_sym (s:=s) (s':=s').
+ Definition eq_trans (s s' s'' : t) :=
+ Raw.eq_trans (s:=s) (s':=s') (s'':=s'').
End Make.
diff --git a/theories/FSets/OrderedType.v b/theories/FSets/OrderedType.v
index 87b7db329..c5edf1de4 100644
--- a/theories/FSets/OrderedType.v
+++ b/theories/FSets/OrderedType.v
@@ -6,13 +6,13 @@
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
-(* $Id$ *)
+(* $Id: OrderedType.v,v 1.4 2006/03/10 10:49:48 letouzey Exp $ *)
Require Export SetoidList.
Set Implicit Arguments.
Unset Strict Implicit.
-(* TODO concernant la tactique order
+(* TODO concernant la tactique order:
* propagate_lt n'est sans doute pas complet
* un propagate_le
* exploiter les hypotheses negatives restant a la fin
@@ -21,26 +21,26 @@ Unset Strict Implicit.
(** * Ordered types *)
-Inductive Compare (X Set) (lt eq : X -> X -> Prop) (x y : X) : Set :=
- | Lt lt x y -> Compare lt eq x y
- | Eq eq x y -> Compare lt eq x y
- | Gt lt y x -> Compare lt eq x y.
+Inductive Compare (X : Set) (lt eq : X -> X -> Prop) (x y : X) : Set :=
+ | Lt : lt x y -> Compare lt eq x y
+ | Eq : eq x y -> Compare lt eq x y
+ | Gt : lt y x -> Compare lt eq x y.
Module Type OrderedType.
- Parameter t Set.
+ Parameter t : Set.
- Parameter eq t -> t -> Prop.
- Parameter lt t -> t -> Prop.
+ Parameter eq : t -> t -> Prop.
+ Parameter lt : t -> t -> Prop.
- Axiom eq_refl forall x : t, eq x x.
- Axiom eq_sym forall x y : t, eq x y -> eq y x.
- Axiom eq_trans forall x y z : t, eq x y -> eq y z -> eq x z.
+ Axiom eq_refl : forall x : t, eq x x.
+ Axiom eq_sym : forall x y : t, eq x y -> eq y x.
+ Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
- Axiom lt_trans forall x y z : t, lt x y -> lt y z -> lt x z.
- Axiom lt_not_eq forall x y : t, lt x y -> ~ eq x y.
+ Axiom lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
+ Axiom lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
- Parameter compare forall x y : t, Compare lt eq x y.
+ Parameter compare : forall x y : t, Compare lt eq x y.
Hint Immediate eq_sym.
Hint Resolve eq_refl eq_trans lt_not_eq lt_trans.
@@ -52,51 +52,51 @@ End OrderedType.
(** Additional properties that can be derived from signature
[OrderedType]. *)
-Module OrderedTypeFacts (O OrderedType).
+Module OrderedTypeFacts (O: OrderedType).
Import O.
- Lemma lt_antirefl forall x, ~ lt x x.
+ Lemma lt_antirefl : forall x, ~ lt x x.
Proof.
intros; intro; absurd (eq x x); auto.
Qed.
- Lemma lt_eq forall x y z, lt x y -> eq y z -> lt x z.
+ Lemma lt_eq : forall x y z, lt x y -> eq y z -> lt x z.
Proof.
intros; destruct (compare x z); auto.
elim (lt_not_eq H); apply eq_trans with z; auto.
elim (lt_not_eq (lt_trans l H)); auto.
Qed.
- Lemma eq_lt forall x y z, eq x y -> lt y z -> lt x z.
+ Lemma eq_lt : forall x y z, eq x y -> lt y z -> lt x z.
Proof.
intros; destruct (compare x z); auto.
elim (lt_not_eq H0); apply eq_trans with x; auto.
elim (lt_not_eq (lt_trans H0 l)); auto.
Qed.
- Lemma le_eq forall x y z, ~lt x y -> eq y z -> ~lt x z.
+ Lemma le_eq : forall x y z, ~lt x y -> eq y z -> ~lt x z.
Proof.
intros; intro; destruct H; apply lt_eq with z; auto.
Qed.
- Lemma eq_le forall x y z, eq x y -> ~lt y z -> ~lt x z.
+ Lemma eq_le : forall x y z, eq x y -> ~lt y z -> ~lt x z.
Proof.
intros; intro; destruct H0; apply eq_lt with x; auto.
Qed.
- Lemma neq_eq forall x y z, ~eq x y -> eq y z -> ~eq x z.
+ Lemma neq_eq : forall x y z, ~eq x y -> eq y z -> ~eq x z.
Proof.
intros; intro; destruct H; apply eq_trans with z; auto.
Qed.
- Lemma eq_neq forall x y z, eq x y -> ~eq y z -> ~eq x z.
+ Lemma eq_neq : forall x y z, eq x y -> ~eq y z -> ~eq x z.
Proof.
intros; intro; destruct H0; apply eq_trans with x; auto.
Qed.
Hint Immediate eq_lt lt_eq le_eq eq_le neq_eq eq_neq.
- Lemma le_lt_trans forall x y z, ~lt y x -> lt y z -> lt x z.
+ Lemma le_lt_trans : forall x y z, ~lt y x -> lt y z -> lt x z.
Proof.
intros; destruct (compare y x); auto.
elim (H l).
@@ -104,7 +104,7 @@ Module OrderedTypeFacts (O OrderedType).
apply lt_trans with y; auto.
Qed.
- Lemma lt_le_trans forall x y z, lt x y -> ~lt z y -> lt x z.
+ Lemma lt_le_trans : forall x y z, lt x y -> ~lt z y -> lt x z.
Proof.
intros; destruct (compare z y); auto.
elim (H0 l).
@@ -112,58 +112,58 @@ Module OrderedTypeFacts (O OrderedType).
apply lt_trans with y; auto.
Qed.
- Lemma le_neq forall x y, ~lt x y -> ~eq x y -> lt y x.
+ Lemma le_neq : forall x y, ~lt x y -> ~eq x y -> lt y x.
Proof.
intros; destruct (compare x y); intuition.
Qed.
- Lemma neq_sym forall x y, ~eq x y -> ~eq y x.
+ Lemma neq_sym : forall x y, ~eq x y -> ~eq y x.
Proof.
intuition.
Qed.
-Ltac abstraction = match goal with
+Ltac abstraction := match goal with
(* First, some obvious simplifications *)
- | H False |- _ => elim H
- | H lt ?x ?x |- _ => elim (lt_antirefl H)
- | H ~eq ?x ?x |- _ => elim (H (eq_refl x))
- | H eq ?x ?x |- _ => clear H; abstraction
- | H ~lt ?x ?x |- _ => clear H; abstraction
+ | H : False |- _ => elim H
+ | H : lt ?x ?x |- _ => elim (lt_antirefl H)
+ | H : ~eq ?x ?x |- _ => elim (H (eq_refl x))
+ | H : eq ?x ?x |- _ => clear H; abstraction
+ | H : ~lt ?x ?x |- _ => clear H; abstraction
| |- eq ?x ?x => exact (eq_refl x)
| |- lt ?x ?x => elimtype False; abstraction
| |- ~ _ => intro; abstraction
- | H1 ~lt ?x ?y, H2: ~eq ?x ?y |- _ =>
+ | H1: ~lt ?x ?y, H2: ~eq ?x ?y |- _ =>
generalize (le_neq H1 H2); clear H1 H2; intro; abstraction
- | H1 ~lt ?x ?y, H2: ~eq ?y ?x |- _ =>
+ | H1: ~lt ?x ?y, H2: ~eq ?y ?x |- _ =>
generalize (le_neq H1 (neq_sym H2)); clear H1 H2; intro; abstraction
(* Then, we generalize all interesting facts *)
- | H lt ?x ?y |- _ => revert H; abstraction
- | H ~lt ?x ?y |- _ => revert H; abstraction
- | H ~eq ?x ?y |- _ => revert H; abstraction
- | H eq ?x ?y |- _ => revert H; abstraction
+ | H : lt ?x ?y |- _ => revert H; abstraction
+ | H : ~lt ?x ?y |- _ => revert H; abstraction
+ | H : ~eq ?x ?y |- _ => revert H; abstraction
+ | H : eq ?x ?y |- _ => revert H; abstraction
| _ => idtac
end.
-Ltac do_eq a b Eq = match goal with
- | |- lt ?x ?y -> _ => let H = fresh "H" in
+Ltac do_eq a b Eq := match goal with
+ | |- lt ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_lt (eq_sym Eq) H); clear H; intro H) ||
(generalize (lt_eq H Eq); clear H; intro H) ||
idtac);
do_eq a b Eq
- | |- ~lt ?x ?y -> _ => let H = fresh "H" in
+ | |- ~lt ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_le (eq_sym Eq) H); clear H; intro H) ||
(generalize (le_eq H Eq); clear H; intro H) ||
idtac);
do_eq a b Eq
- | |- eq ?x ?y -> _ => let H = fresh "H" in
+ | |- eq ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_trans (eq_sym Eq) H); clear H; intro H) ||
(generalize (eq_trans H Eq); clear H; intro H) ||
idtac);
do_eq a b Eq
- | |- ~eq ?x ?y -> _ => let H = fresh "H" in
+ | |- ~eq ?x ?y -> _ => let H := fresh "H" in
(intro H;
(generalize (eq_neq (eq_sym Eq) H); clear H; intro H) ||
(generalize (neq_eq H Eq); clear H; intro H) ||
@@ -176,44 +176,44 @@ Ltac do_eq a b Eq = match goal with
| _ => idtac
end.
-Ltac propagate_eq = abstraction; clear; match goal with
+Ltac propagate_eq := abstraction; clear; match goal with
(* the abstraction tactic leaves equality facts in head position...*)
| |- eq ?a ?b -> _ =>
- let Eq = fresh "Eq" in (intro Eq; do_eq a b Eq; clear Eq);
+ let Eq := fresh "Eq" in (intro Eq; do_eq a b Eq; clear Eq);
propagate_eq
| _ => idtac
end.
-Ltac do_lt x y Lt = match goal with
+Ltac do_lt x y Lt := match goal with
(* Lt *)
| |- lt x y -> _ => intros _; do_lt x y Lt
- | |- lt y ?z -> _ => let H = fresh "H" in
+ | |- lt y ?z -> _ => let H := fresh "H" in
(intro H; generalize (lt_trans Lt H); intro); do_lt x y Lt
- | |- lt ?z x -> _ => let H = fresh "H" in
+ | |- lt ?z x -> _ => let H := fresh "H" in
(intro H; generalize (lt_trans H Lt); intro); do_lt x y Lt
| |- lt _ _ -> _ => intro; do_lt x y Lt
(* Ge *)
| |- ~lt y x -> _ => intros _; do_lt x y Lt
- | |- ~lt x ?z -> _ => let H = fresh "H" in
+ | |- ~lt x ?z -> _ => let H := fresh "H" in
(intro H; generalize (le_lt_trans H Lt); intro); do_lt x y Lt
- | |- ~lt ?z y -> _ => let H = fresh "H" in
+ | |- ~lt ?z y -> _ => let H := fresh "H" in
(intro H; generalize (lt_le_trans Lt H); intro); do_lt x y Lt
| |- ~lt _ _ -> _ => intro; do_lt x y Lt
| _ => idtac
end.
-Definition hide_lt = lt.
+Definition hide_lt := lt.
-Ltac propagate_lt = abstraction; match goal with
+Ltac propagate_lt := abstraction; match goal with
(* when no [=] remains, the abstraction tactic leaves [<] facts first. *)
| |- lt ?x ?y -> _ =>
- let Lt = fresh "Lt" in (intro Lt; do_lt x y Lt;
+ let Lt := fresh "Lt" in (intro Lt; do_lt x y Lt;
change (hide_lt x y) in Lt);
propagate_lt
| _ => unfold hide_lt in *
end.
-Ltac order =
+Ltac order :=
intros;
propagate_eq;
propagate_lt;
@@ -221,91 +221,91 @@ Ltac order =
propagate_lt;
eauto.
-Ltac false_order = elimtype False; order.
+Ltac false_order := elimtype False; order.
- Lemma gt_not_eq forall x y, lt y x -> ~ eq x y.
+ Lemma gt_not_eq : forall x y, lt y x -> ~ eq x y.
Proof.
order.
Qed.
- Lemma eq_not_lt forall x y : t, eq x y -> ~ lt x y.
+ Lemma eq_not_lt : forall x y : t, eq x y -> ~ lt x y.
Proof.
order.
Qed.
Hint Resolve gt_not_eq eq_not_lt.
- Lemma eq_not_gt forall x y : t, eq x y -> ~ lt y x.
+ Lemma eq_not_gt : forall x y : t, eq x y -> ~ lt y x.
Proof.
order.
Qed.
- Lemma lt_not_gt forall x y : t, lt x y -> ~ lt y x.
+ Lemma lt_not_gt : forall x y : t, lt x y -> ~ lt y x.
Proof.
order.
Qed.
Hint Resolve eq_not_gt lt_antirefl lt_not_gt.
- Lemma elim_compare_eq
- forall x y t,
- eq x y -> exists H eq x y, compare x y = Eq _ H.
+ Lemma elim_compare_eq :
+ forall x y : t,
+ eq x y -> exists H : eq x y, compare x y = Eq _ H.
Proof.
intros; case (compare x y); intros H'; try solve [false_order].
exists H'; auto.
Qed.
- Lemma elim_compare_lt
- forall x y t,
- lt x y -> exists H lt x y, compare x y = Lt _ H.
+ Lemma elim_compare_lt :
+ forall x y : t,
+ lt x y -> exists H : lt x y, compare x y = Lt _ H.
Proof.
intros; case (compare x y); intros H'; try solve [false_order].
exists H'; auto.
Qed.
- Lemma elim_compare_gt
- forall x y t,
- lt y x -> exists H lt y x, compare x y = Gt _ H.
+ Lemma elim_compare_gt :
+ forall x y : t,
+ lt y x -> exists H : lt y x, compare x y = Gt _ H.
Proof.
intros; case (compare x y); intros H'; try solve [false_order].
exists H'; auto.
Qed.
- Ltac elim_comp =
+ Ltac elim_comp :=
match goal with
| |- ?e => match e with
| context ctx [ compare ?a ?b ] =>
- let H = fresh in
+ let H := fresh in
(destruct (compare a b) as [H|H|H];
try solve [ intros; false_order])
end
end.
- Ltac elim_comp_eq x y =
- elim (elim_compare_eq (x=x) (y:=y));
+ Ltac elim_comp_eq x y :=
+ elim (elim_compare_eq (x:=x) (y:=y));
[ intros _1 _2; rewrite _2; clear _1 _2 | auto ].
- Ltac elim_comp_lt x y =
- elim (elim_compare_lt (x=x) (y:=y));
+ Ltac elim_comp_lt x y :=
+ elim (elim_compare_lt (x:=x) (y:=y));
[ intros _1 _2; rewrite _2; clear _1 _2 | auto ].
- Ltac elim_comp_gt x y =
- elim (elim_compare_gt (x=x) (y:=y));
+ Ltac elim_comp_gt x y :=
+ elim (elim_compare_gt (x:=x) (y:=y));
[ intros _1 _2; rewrite _2; clear _1 _2 | auto ].
- Lemma eq_dec forall x y : t, {eq x y} + {~ eq x y}.
+ Lemma eq_dec : forall x y : t, {eq x y} + {~ eq x y}.
Proof.
intros; elim (compare x y); [ right | left | right ]; auto.
Qed.
- Lemma lt_dec forall x y : t, {lt x y} + {~ lt x y}.
+ Lemma lt_dec : forall x y : t, {lt x y} + {~ lt x y}.
Proof.
intros; elim (compare x y); [ left | right | right ]; auto.
Qed.
- Definition eqb x y bool := if eq_dec x y then true else false.
+ Definition eqb x y : bool := if eq_dec x y then true else false.
- Lemma eqb_alt
+ Lemma eqb_alt :
forall x y, eqb x y = match compare x y with Eq _ => true | _ => false end.
Proof.
unfold eqb; intros; destruct (eq_dec x y); elim_comp; auto.
@@ -313,37 +313,37 @@ Ltac false_order = elimtype False; order.
(* Specialization of resuts about lists modulo. *)
-Notation In=(InA eq).
-Notation Inf=(lelistA lt).
-Notation Sort=(sort lt).
-Notation NoRedun=(noredunA eq).
+Notation In:=(InA eq).
+Notation Inf:=(lelistA lt).
+Notation Sort:=(sort lt).
+Notation NoRedun:=(noredunA eq).
-Lemma In_eq forall l x y, eq x y -> In x l -> In y l.
+Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.
Proof. exact (InA_eqA eq_sym eq_trans). Qed.
-Lemma ListIn_In forall l x, List.In x l -> In x l.
+Lemma ListIn_In : forall l x, List.In x l -> In x l.
Proof. exact (In_InA eq_refl). Qed.
-Lemma Inf_lt forall l x y, lt x y -> Inf y l -> Inf x l.
+Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l.
Proof. exact (InfA_ltA lt_trans). Qed.
-Lemma Inf_eq forall l x y, eq x y -> Inf y l -> Inf x l.
+Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l.
Proof. exact (InfA_eqA eq_lt). Qed.
-Lemma Sort_Inf_In forall l x a, Sort l -> Inf a l -> In x l -> lt a x.
+Lemma Sort_Inf_In : forall l x a, Sort l -> Inf a l -> In x l -> lt a x.
Proof. exact (SortA_InfA_InA eq_refl eq_sym lt_trans lt_eq eq_lt). Qed.
-Lemma ListIn_Inf forall l x, (forall y, List.In y l -> lt x y) -> Inf x l.
+Lemma ListIn_Inf : forall l x, (forall y, List.In y l -> lt x y) -> Inf x l.
Proof. exact (@In_InfA t lt). Qed.
-Lemma In_Inf forall l x, (forall y, In y l -> lt x y) -> Inf x l.
-Proof. exact (InA_InfA eq_refl (ltA=lt)). Qed.
+Lemma In_Inf : forall l x, (forall y, In y l -> lt x y) -> Inf x l.
+Proof. exact (InA_InfA eq_refl (ltA:=lt)). Qed.
-Lemma Inf_alt
+Lemma Inf_alt :
forall l x, Sort l -> (Inf x l <-> (forall y, In y l -> lt x y)).
Proof. exact (InfA_alt eq_refl eq_sym lt_trans lt_eq eq_lt). Qed.
-Lemma Sort_NoRedun forall l, Sort l -> NoRedun l.
+Lemma Sort_NoRedun : forall l, Sort l -> NoRedun l.
Proof. exact (SortA_noredunA eq_refl eq_sym lt_trans lt_not_eq lt_eq eq_lt) . Qed.
Hint Resolve ListIn_In Sort_NoRedun Inf_lt.
@@ -351,68 +351,68 @@ Hint Immediate In_eq Inf_lt.
End OrderedTypeFacts.
-Module PairOrderedType(OOrderedType).
+Module PairOrderedType(O:OrderedType).
Import O.
- Module MO=OrderedTypeFacts(O).
+ Module MO:=OrderedTypeFacts(O).
Import MO.
Section Elt.
- Variable elt Set.
- Notation key=t.
+ Variable elt : Set.
+ Notation key:=t.
- Definition eqk (p p'key*elt) := eq (fst p) (fst p').
- Definition eqke (p p'key*elt) :=
+ Definition eqk (p p':key*elt) := eq (fst p) (fst p').
+ Definition eqke (p p':key*elt) :=
eq (fst p) (fst p') /\ (snd p) = (snd p').
- Definition ltk (p p'key*elt) := lt (fst p) (fst p').
+ Definition ltk (p p':key*elt) := lt (fst p) (fst p').
Hint Unfold eqk eqke ltk.
Hint Extern 2 (eqke ?a ?b) => split.
(* eqke is stricter than eqk *)
- Lemma eqke_eqk forall x x', eqke x x' -> eqk x x'.
+ Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.
Proof.
unfold eqk, eqke; intuition.
Qed.
(* ltk ignore the second components *)
- Lemma ltk_right_r forall x k e e', ltk x (k,e) -> ltk x (k,e').
+ Lemma ltk_right_r : forall x k e e', ltk x (k,e) -> ltk x (k,e').
Proof. auto. Qed.
- Lemma ltk_right_l forall x k e e', ltk (k,e) x -> ltk (k,e') x.
+ Lemma ltk_right_l : forall x k e e', ltk (k,e) x -> ltk (k,e') x.
Proof. auto. Qed.
Hint Immediate ltk_right_r ltk_right_l.
(* eqk, eqke are equalities, ltk is a strict order *)
- Lemma eqk_refl forall e, eqk e e.
+ Lemma eqk_refl : forall e, eqk e e.
Proof. auto. Qed.
- Lemma eqke_refl forall e, eqke e e.
+ Lemma eqke_refl : forall e, eqke e e.
Proof. auto. Qed.
- Lemma eqk_sym forall e e', eqk e e' -> eqk e' e.
+ Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e.
Proof. auto. Qed.
- Lemma eqke_sym forall e e', eqke e e' -> eqke e' e.
+ Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e.
Proof. unfold eqke; intuition. Qed.
- Lemma eqk_trans forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.
+ Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.
Proof. eauto. Qed.
- Lemma eqke_trans forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.
+ Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.
Proof.
unfold eqke; intuition; [ eauto | congruence ].
Qed.
- Lemma ltk_trans forall e e' e'', ltk e e' -> ltk e' e'' -> ltk e e''.
+ Lemma ltk_trans : forall e e' e'', ltk e e' -> ltk e' e'' -> ltk e e''.
Proof. eauto. Qed.
- Lemma ltk_not_eqk forall e e', ltk e e' -> ~ eqk e e'.
+ Lemma ltk_not_eqk : forall e e', ltk e e' -> ~ eqk e e'.
Proof. unfold eqk, ltk; auto. Qed.
- Lemma ltk_not_eqke forall e e', ltk e e' -> ~eqke e e'.
+ Lemma ltk_not_eqke : forall e e', ltk e e' -> ~eqke e e'.
Proof.
unfold eqke, ltk; intuition; simpl in *; subst.
exact (lt_not_eq H H1).
@@ -424,15 +424,15 @@ Module PairOrderedType(OOrderedType).
(* Additionnal facts *)
- Lemma eqk_not_ltk forall x x', eqk x x' -> ~ltk x x'.
+ Lemma eqk_not_ltk : forall x x', eqk x x' -> ~ltk x x'.
Proof.
unfold eqk, ltk; simpl; auto.
Qed.
- Lemma ltk_eqk forall e e' e'', ltk e e' -> eqk e' e'' -> ltk e e''.
+ Lemma ltk_eqk : forall e e' e'', ltk e e' -> eqk e' e'' -> ltk e e''.
Proof. eauto. Qed.
- Lemma eqk_ltk forall e e' e'', eqk e e' -> ltk e' e'' -> ltk e e''.
+ Lemma eqk_ltk : forall e e' e'', eqk e e' -> ltk e' e'' -> ltk e e''.
Proof.
intros (k,e) (k',e') (k'',e'').
unfold ltk, eqk; simpl; eauto.
@@ -440,23 +440,23 @@ Module PairOrderedType(OOrderedType).
Hint Resolve eqk_not_ltk.
Hint Immediate ltk_eqk eqk_ltk.
- Lemma InA_eqke_eqk
+ Lemma InA_eqke_eqk :
forall x m, InA eqke x m -> InA eqk x m.
Proof.
unfold eqke; induction 1; intuition.
Qed.
Hint Resolve InA_eqke_eqk.
- Definition MapsTo (kkey)(e:elt):= InA eqke (k,e).
- Definition In k m = exists e:elt, MapsTo k e m.
- Notation Sort = (sort ltk).
- Notation Inf = (lelistA ltk).
+ Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
+ Definition In k m := exists e:elt, MapsTo k e m.
+ Notation Sort := (sort ltk).
+ Notation Inf := (lelistA ltk).
Hint Unfold MapsTo In.
(* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *)
- Lemma In_alt forall k l, In k l <-> exists e, InA eqk (k,e) l.
+ Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l.
Proof.
firstorder.
exists x; auto.
@@ -467,32 +467,32 @@ Module PairOrderedType(OOrderedType).
exists e; auto.
Qed.
- Lemma MapsTo_eq forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
+ Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
Proof.
intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto.
Qed.
- Lemma In_eq forall l x y, eq x y -> In x l -> In y l.
+ Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.
Proof.
destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto.
Qed.
- Lemma Inf_eq forall l x x', eqk x x' -> Inf x' l -> Inf x l.
+ Lemma Inf_eq : forall l x x', eqk x x' -> Inf x' l -> Inf x l.
Proof. exact (InfA_eqA eqk_ltk). Qed.
- Lemma Inf_lt forall l x x', ltk x x' -> Inf x' l -> Inf x l.
+ Lemma Inf_lt : forall l x x', ltk x x' -> Inf x' l -> Inf x l.
Proof. exact (InfA_ltA ltk_trans). Qed.
Hint Immediate Inf_eq.
Hint Resolve Inf_lt.
- Lemma Sort_Inf_In
+ Lemma Sort_Inf_In :
forall l p q, Sort l -> Inf q l -> InA eqk p l -> ltk q p.
Proof.
exact (SortA_InfA_InA eqk_refl eqk_sym ltk_trans ltk_eqk eqk_ltk).
Qed.
- Lemma Sort_Inf_NotIn
+ Lemma Sort_Inf_NotIn :
forall l k e, Sort l -> Inf (k,e) l -> ~In k l.
Proof.
intros; red; intros.
@@ -502,45 +502,45 @@ Module PairOrderedType(OOrderedType).
red; simpl; auto.
Qed.
- Lemma Sort_noredunA forall l, Sort l -> noredunA eqk l.
+ Lemma Sort_noredunA: forall l, Sort l -> noredunA eqk l.
Proof.
exact (SortA_noredunA eqk_refl eqk_sym ltk_trans ltk_not_eqk ltk_eqk eqk_ltk).
Qed.
- Lemma Sort_In_cons_1 forall e l e', Sort (e::l) -> InA eqk e' l -> ltk e e'.
+ Lemma Sort_In_cons_1 : forall e l e', Sort (e::l) -> InA eqk e' l -> ltk e e'.
Proof.
inversion 1; intros; eapply Sort_Inf_In; eauto.
Qed.
- Lemma Sort_In_cons_2 forall l e e', Sort (e::l) -> InA eqk e' (e::l) ->
+ Lemma Sort_In_cons_2 : forall l e e', Sort (e::l) -> InA eqk e' (e::l) ->
ltk e e' \/ eqk e e'.
Proof.
inversion_clear 2; auto.
left; apply Sort_In_cons_1 with l; auto.
Qed.
- Lemma Sort_In_cons_3
- forall x l k e, Sort ((k,e):l) -> In x l -> ~eq x k.
+ Lemma Sort_In_cons_3 :
+ forall x l k e, Sort ((k,e)::l) -> In x l -> ~eq x k.
Proof.
inversion_clear 1; red; intros.
destruct (Sort_Inf_NotIn H0 H1 (In_eq H2 H)).
Qed.
- Lemma In_inv forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
+ Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
Proof.
inversion 1.
inversion_clear H0; eauto.
destruct H1; simpl in *; intuition.
Qed.
- Lemma In_inv_2 forall k k' e e' l,
- InA eqk (k, e) ((k', e') : l) -> ~ eq k k' -> InA eqk (k, e) l.
+ Lemma In_inv_2 : forall k k' e e' l,
+ InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l.
Proof.
inversion_clear 1; compute in H0; intuition.
Qed.
- Lemma In_inv_3 forall x x' l,
- InA eqke x (x' : l) -> ~ eqk x x' -> InA eqke x l.
+ Lemma In_inv_3 : forall x x' l,
+ InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
Proof.
inversion_clear 1; compute in H0; intuition.
Qed.
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-26 18:32:02 +0000