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-rw-r--r--theories/MSets/MSetPositive.v62
1 files changed, 27 insertions, 35 deletions
diff --git a/theories/MSets/MSetPositive.v b/theories/MSets/MSetPositive.v
index e500602f..25a8c162 100644
--- a/theories/MSets/MSetPositive.v
+++ b/theories/MSets/MSetPositive.v
@@ -19,14 +19,9 @@
Require Import Bool BinPos Orders MSetInterface.
Set Implicit Arguments.
-
Local Open Scope lazy_bool_scope.
Local Open Scope positive_scope.
-
Local Unset Elimination Schemes.
-Local Unset Case Analysis Schemes.
-Local Unset Boolean Equality Schemes.
-
(** Even if [positive] can be seen as an ordered type with respect to the
usual order (see above), we can also use a lexicographic order over bits
@@ -98,7 +93,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
Module E:=PositiveOrderedTypeBits.
- Definition elt := positive.
+ Definition elt := positive : Type.
Inductive tree :=
| Leaf : tree
@@ -106,9 +101,9 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
Scheme tree_ind := Induction for tree Sort Prop.
- Definition t := tree.
+ Definition t := tree : Type.
- Definition empty := Leaf.
+ Definition empty : t := Leaf.
Fixpoint is_empty (m : t) : bool :=
match m with
@@ -116,7 +111,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
| Node l b r => negb b &&& is_empty l &&& is_empty r
end.
- Fixpoint mem (i : positive) (m : t) : bool :=
+ Fixpoint mem (i : positive) (m : t) {struct m} : bool :=
match m with
| Leaf => false
| Node l o r =>
@@ -147,13 +142,13 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
(** helper function to avoid creating empty trees that are not leaves *)
- Definition node l (b: bool) r :=
+ Definition node (l : t) (b: bool) (r : t) : t :=
if b then Node l b r else
match l,r with
| Leaf,Leaf => Leaf
| _,_ => Node l false r end.
- Fixpoint remove (i : positive) (m : t) : t :=
+ Fixpoint remove (i : positive) (m : t) {struct m} : t :=
match m with
| Leaf => Leaf
| Node l o r =>
@@ -164,7 +159,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
end
end.
- Fixpoint union (m m': t) :=
+ Fixpoint union (m m': t) : t :=
match m with
| Leaf => m'
| Node l o r =>
@@ -174,7 +169,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
end
end.
- Fixpoint inter (m m': t) :=
+ Fixpoint inter (m m': t) : t :=
match m with
| Leaf => Leaf
| Node l o r =>
@@ -184,7 +179,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
end
end.
- Fixpoint diff (m m': t) :=
+ Fixpoint diff (m m': t) : t :=
match m with
| Leaf => Leaf
| Node l o r =>
@@ -216,7 +211,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
(** reverses [y] and concatenate it with [x] *)
- Fixpoint rev_append y x :=
+ Fixpoint rev_append (y x : elt) : elt :=
match y with
| 1 => x
| y~1 => rev_append y x~1
@@ -267,14 +262,14 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
end.
Definition exists_ m := xexists m 1.
- Fixpoint xfilter (m : t) (i : positive) :=
+ Fixpoint xfilter (m : t) (i : positive) : t :=
match m with
| Leaf => Leaf
| Node l o r => node (xfilter l i~0) (o &&& f (rev i)) (xfilter r i~1)
end.
Definition filter m := xfilter m 1.
- Fixpoint xpartition (m : t) (i : positive) :=
+ Fixpoint xpartition (m : t) (i : positive) : t * t :=
match m with
| Leaf => (Leaf,Leaf)
| Node l o r =>
@@ -316,7 +311,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
(** would it be more efficient to use a path like in the above functions ? *)
- Fixpoint choose (m: t) :=
+ Fixpoint choose (m: t) : option elt :=
match m with
| Leaf => None
| Node l o r => if o then Some 1 else
@@ -326,7 +321,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
end
end.
- Fixpoint min_elt (m: t) :=
+ Fixpoint min_elt (m: t) : option elt :=
match m with
| Leaf => None
| Node l o r =>
@@ -336,7 +331,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
end
end.
- Fixpoint max_elt (m: t) :=
+ Fixpoint max_elt (m: t) : option elt :=
match m with
| Leaf => None
| Node l o r =>
@@ -414,10 +409,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
case o; trivial.
destruct l; trivial.
destruct r; trivial.
- symmetry. destruct x.
- apply mem_Leaf.
- apply mem_Leaf.
- reflexivity.
+ destruct x; reflexivity.
Qed.
Local Opaque node.
@@ -427,7 +419,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
Proof.
unfold Empty, In.
induction s as [|l IHl o r IHr]; simpl.
- setoid_rewrite mem_Leaf. firstorder.
+ firstorder.
rewrite <- 2andb_lazy_alt, 2andb_true_iff, IHl, IHr. clear IHl IHr.
destruct o; simpl; split.
intuition discriminate.
@@ -813,7 +805,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
rewrite <- andb_lazy_alt. apply andb_true_iff.
Qed.
- Lemma filter_spec: forall s x f, compat_bool E.eq f ->
+ Lemma filter_spec: forall s x f, @compat_bool elt E.eq f ->
(In x (filter f s) <-> In x s /\ f x = true).
Proof. intros. apply xfilter_spec. Qed.
@@ -824,7 +816,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
Proof.
unfold For_all, In. intro f.
induction s as [|l IHl o r IHr]; intros i; simpl.
- setoid_rewrite mem_Leaf. intuition discriminate.
+ intuition discriminate.
rewrite <- 2andb_lazy_alt, <- orb_lazy_alt, 2 andb_true_iff.
rewrite IHl, IHr. clear IHl IHr.
split.
@@ -838,7 +830,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
apply H. assumption.
Qed.
- Lemma for_all_spec: forall s f, compat_bool E.eq f ->
+ Lemma for_all_spec: forall s f, @compat_bool elt E.eq f ->
(for_all f s = true <-> For_all (fun x => f x = true) s).
Proof. intros. apply xforall_spec. Qed.
@@ -849,7 +841,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
Proof.
unfold Exists, In. intro f.
induction s as [|l IHl o r IHr]; intros i; simpl.
- setoid_rewrite mem_Leaf. firstorder.
+ firstorder.
rewrite <- 2orb_lazy_alt, 2orb_true_iff, <- andb_lazy_alt, andb_true_iff.
rewrite IHl, IHr. clear IHl IHr.
split.
@@ -860,7 +852,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
intros [[x|x|] H]; eauto.
Qed.
- Lemma exists_spec : forall s f, compat_bool E.eq f ->
+ Lemma exists_spec : forall s f, @compat_bool elt E.eq f ->
(exists_ f s = true <-> Exists (fun x => f x = true) s).
Proof. intros. apply xexists_spec. Qed.
@@ -876,11 +868,11 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
destruct o; simpl; rewrite IHl, IHr; reflexivity.
Qed.
- Lemma partition_spec1 : forall s f, compat_bool E.eq f ->
+ Lemma partition_spec1 : forall s f, @compat_bool elt E.eq f ->
Equal (fst (partition f s)) (filter f s).
Proof. intros. rewrite partition_filter. reflexivity. Qed.
- Lemma partition_spec2 : forall s f, compat_bool E.eq f ->
+ Lemma partition_spec2 : forall s f, @compat_bool elt E.eq f ->
Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
Proof. intros. rewrite partition_filter. reflexivity. Qed.
@@ -897,7 +889,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
induction s as [|l IHl o r IHr]; simpl.
intros. split; intro H.
left. assumption.
- destruct H as [H|[x [Hx Hx']]]. assumption. elim (empty_spec Hx').
+ destruct H as [H|[x [Hx Hx']]]. assumption. discriminate.
intros j acc y. case o.
rewrite IHl. rewrite InA_cons. rewrite IHr. clear IHl IHr. split.
@@ -1087,7 +1079,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
destruct (min_elt r).
injection H. intros <-. clear H.
destruct y as [z|z|].
- apply (IHr p z); trivial.
+ apply (IHr e z); trivial.
elim (Hp _ H').
discriminate.
discriminate.
@@ -1141,7 +1133,7 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits.
injection H. intros <-. clear H.
destruct y as [z|z|].
elim (Hp _ H').
- apply (IHl p z); trivial.
+ apply (IHl e z); trivial.
discriminate.
discriminate.
Qed.
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-17 02:06:24 +0000