8 files changed, 22 insertions, 59 deletions
diff --git a/theories/Init/Logic.v b/theories/Init/Logic.v
index 14832e0b7..97f3b126c 100644
--- a/theories/Init/Logic.v
+++ b/theories/Init/Logic.v
@@ -216,11 +216,9 @@ Definition all (A:Type) (P:A -> Prop) := forall x:A, P x.
 
 (* Rule order is important to give printing priority to fully typed exists *)
 
-Notation "'exists' x , p" := (ex (fun x => p))
-  (at level 200, x ident, right associativity) : type_scope.
-Notation "'exists' x : t , p" := (ex (fun x:t => p))
-  (at level 200, x ident, right associativity,
-    format "'[' 'exists'  '/  ' x  :  t ,  '/  ' p ']'")
+Notation "'exists' x .. y , p" := (ex (fun x => .. (ex (fun y => p)) ..))
+  (at level 200, x binder, right associativity,
+   format "'[' 'exists'  '/  ' x .. y ,  '/  ' p ']'")
   : type_scope.
 
 Notation "'exists2' x , p & q" := (ex2 (fun x => p) (fun x => q))
@@ -390,13 +388,11 @@ Definition uniqueness (A:Type) (P:A->Prop) := forall x y, P x -> P y -> x = y.
 
 (** Unique existence *)
 
-Notation "'exists' ! x , P" := (ex (unique (fun x => P)))
-  (at level 200, x ident, right associativity,
-    format "'[' 'exists' !  '/  ' x ,  '/  ' P ']'") : type_scope.
-Notation "'exists' ! x : A , P" :=
-  (ex (unique (fun x:A => P)))
-  (at level 200, x ident, right associativity,
-    format "'[' 'exists' !  '/  ' x  :  A ,  '/  ' P ']'") : type_scope.
+Notation "'exists' ! x .. y , p" :=
+  (ex (unique (fun x => .. (ex (unique (fun y => p))) ..)))
+  (at level 200, x binder, right associativity,
+   format "'[' 'exists'  !  '/  ' x .. y ,  '/  ' p ']'")
+  : type_scope.
 
 Lemma unique_existence : forall (A:Type) (P:A->Prop),
   ((exists x, P x) /\ uniqueness P) <-> (exists! x, P x).
diff --git a/theories/Lists/List.v b/theories/Lists/List.v
index d1e3f90b3..25f96292d 100644
--- a/theories/Lists/List.v
+++ b/theories/Lists/List.v
@@ -117,7 +117,7 @@ Section Facts.
     unfold not; intros a H; inversion_clear H.
   Qed.
 
-  Theorem in_split : forall x (l:list A), In x l -> exists l1, exists l2, l = l1++x::l2.
+  Theorem in_split : forall x (l:list A), In x l -> exists l1 l2, l = l1++x::l2.
   Proof.
   induction l; simpl; destruct 1.
   subst a; auto.
@@ -1641,7 +1641,7 @@ Proof. exact Forall2_nil. Qed.
 
 Theorem Forall2_app_inv_l : forall A B (R:A->B->Prop) l1 l2 l',
   Forall2 R (l1 ++ l2) l' ->
-  exists l1', exists l2', Forall2 R l1 l1' /\ Forall2 R l2 l2' /\ l' = l1' ++ l2'.
+  exists l1' l2', Forall2 R l1 l1' /\ Forall2 R l2 l2' /\ l' = l1' ++ l2'.
 Proof.
   induction l1; intros.
     exists [], l'; auto.
diff --git a/theories/Lists/SetoidList.v b/theories/Lists/SetoidList.v
index 88ccdb126..43a965cf1 100644
--- a/theories/Lists/SetoidList.v
+++ b/theories/Lists/SetoidList.v
@@ -157,8 +157,7 @@ Qed.
 Hint Resolve In_InA.
 
 Lemma InA_split : forall l x, InA x l ->
- exists l1, exists y, exists l2,
- eqA x y /\ l = l1++y::l2.
+ exists l1 y l2, eqA x y /\ l = l1++y::l2.
 Proof.
 induction l; intros; inv.
 exists (@nil A); exists a; exists l; auto.
diff --git a/theories/Logic/JMeq.v b/theories/Logic/JMeq.v
index f17e403ed..330ee79ef 100644
--- a/theories/Logic/JMeq.v
+++ b/theories/Logic/JMeq.v
@@ -111,8 +111,7 @@ apply JMeq_refl.
 Qed.
 
 Lemma eq_dep_strictly_stronger_JMeq :
- exists U, exists P, exists p, exists q, exists x, exists y,
-  JMeq x y /\ ~ eq_dep U P p x q y.
+ exists U P p q x y, JMeq x y /\ ~ eq_dep U P p x q y.
 Proof.
 exists bool. exists (fun _ => True). exists true. exists false.
 exists I. exists I.
diff --git a/theories/MSets/MSetAVL.v b/theories/MSets/MSetAVL.v
index 6825e6881..349cdedf7 100644
--- a/theories/MSets/MSetAVL.v
+++ b/theories/MSets/MSetAVL.v
@@ -1689,7 +1689,7 @@ Proof.
 Qed.
 
 Definition lt (s1 s2 : t) : Prop :=
- exists s1', exists s2', Ok s1' /\ Ok s2' /\ eq s1 s1' /\ eq s2 s2'
+ exists s1' s2', Ok s1' /\ Ok s2' /\ eq s1 s1' /\ eq s2 s2'
    /\ L.lt (elements s1') (elements s2').
 
 Instance lt_strorder : StrictOrder lt.
diff --git a/theories/MSets/MSetList.v b/theories/MSets/MSetList.v
index 45278eaf6..bcf68f1d4 100644
--- a/theories/MSets/MSetList.v
+++ b/theories/MSets/MSetList.v
@@ -786,8 +786,7 @@ Module MakeRaw (X: OrderedType) <: RawSets X.
   Definition eq := L.eq.
   Definition eq_equiv := L.eq_equiv.
   Definition lt l1 l2 :=
-    exists l1', exists l2', Ok l1' /\ Ok l2' /\
-      eq l1 l1' /\ eq l2 l2' /\ L.lt l1' l2'.
+    exists l1' l2', Ok l1' /\ Ok l2' /\ eq l1 l1' /\ eq l2 l2' /\ L.lt l1' l2'.
 
   Instance lt_strorder : StrictOrder lt.
   Proof.
diff --git a/theories/Program/Syntax.v b/theories/Program/Syntax.v
index 2c5fb79c5..bb52a0103 100644
--- a/theories/Program/Syntax.v
+++ b/theories/Program/Syntax.v
@@ -51,15 +51,6 @@ Implicit Arguments Vcons [[A] [n]].
 
 (** Treating n-ary exists *)
 
-Notation " 'exists' x y , p" := (ex (fun x => (ex (fun y => p))))
-  (at level 200, x ident, y ident, right associativity) : type_scope.
-
-Notation " 'exists' x y z , p" := (ex (fun x => (ex (fun y => (ex (fun z => p))))))
-  (at level 200, x ident, y ident, z ident, right associativity) : type_scope.
-
-Notation " 'exists' x y z w , p" := (ex (fun x => (ex (fun y => (ex (fun z => (ex (fun w => p))))))))
-  (at level 200, x ident, y ident, z ident, w ident, right associativity) : type_scope.
-
 Tactic Notation "exists" constr(x) := exists x.
 Tactic Notation "exists" constr(x) constr(y) := exists x ; exists y.
 Tactic Notation "exists" constr(x) constr(y) constr(z) := exists x ; exists y ; exists z.
diff --git a/theories/Unicode/Utf8.v b/theories/Unicode/Utf8.v
index 3a11c9e5e..f806336ba 100644
--- a/theories/Unicode/Utf8.v
+++ b/theories/Unicode/Utf8.v
@@ -8,29 +8,10 @@
 (************************************************************************)
 
 (* Logic *)
-Notation "∀ x , P" := (forall x , P)
-  (at level 200, x ident, right associativity) : type_scope.
-Notation "∀ x y , P" := (forall x y , P)
-  (at level 200, x ident, y ident, right associativity) : type_scope.
-Notation "∀ x y z , P" := (forall x y z , P)
-  (at level 200, x ident, y ident, z ident, right associativity) : type_scope.
-Notation "∀ x y z u , P" := (forall x y z u , P)
-  (at level 200, x ident, y ident, z ident, u ident, right associativity)
-  : type_scope.
-Notation "∀ x : t , P" := (forall x : t , P)
-  (at level 200, x ident, right associativity) : type_scope.
-Notation "∀ x y : t , P" := (forall x y : t , P)
-  (at level 200, x ident, y ident, right associativity) : type_scope.
-Notation "∀ x y z : t , P" := (forall x y z : t , P)
-  (at level 200, x ident, y ident, z ident, right associativity) : type_scope.
-Notation "∀ x y z u : t , P" := (forall x y z u : t , P)
-  (at level 200, x ident, y ident, z ident, u ident, right associativity)
-  : type_scope.
-
-Notation "∃ x , P" := (exists x , P)
-  (at level 200, x ident, right associativity) : type_scope.
-Notation "∃ x : t , P" := (exists x : t, P)
-  (at level 200, x ident, right associativity) : type_scope.
+Notation "∀  x .. y , P" := (forall x, .. (forall y, P) ..)
+  (at level 200, x binder, y binder, right associativity) : type_scope.
+Notation "∃  x .. y , P" := (exists x, .. (exists y, P) ..)
+  (at level 200, x binder, y binder, right associativity) : type_scope.
 
 Notation "x ∨ y" := (x \/ y) (at level 85, right associativity) : type_scope.
 Notation "x ∧ y" := (x /\ y) (at level 80, right associativity) : type_scope.
@@ -40,10 +21,8 @@ Notation "¬ x" := (~x) (at level 75, right associativity) : type_scope.
 Notation "x ≠ y" := (x <> y) (at level 70) : type_scope.
 
 (* Abstraction *)
-(* Not nice
-Notation  "'λ' x : T , y" := ([x:T] y) (at level 1, x,T,y at level 10).
-Notation  "'λ' x := T , y" := ([x:=T] y) (at level 1, x,T,y at level 10).
-*)
+Notation "'λ'  x .. y , t" := (fun x => .. (fun y => t) ..)
+  (at level 200, x binder, y binder, right associativity).
 
 (* Arithmetic *)
 Notation "x ≤ y" := (le x y) (at level 70, no associativity).
@@ -51,10 +30,10 @@ Notation "x ≥ y" := (ge x y) (at level 70, no associativity).
 
 (* test *)
 (*
-Goal ∀ x, True -> (∃ y , x ≥ y + 1) ∨ x ≤ 0.
+Check ∀ x z, True -> (∃ y v, x + v ≥ y + z) ∨ x ≤ 0.
 *)
 
 (* Integer Arithmetic *)
 (* TODO: this should come after ZArith
-Notation "x ≤ y" := (Zle x y) (at level 1, y at level 10).
+Notation "x ≤ y" := (Zle x y) (at level 70, no associativity).
 *)