diff --git a/theories/base.v b/theories/base.v
index 323dd30d14e43c84bb6540f63f5ba17ccc7ceb56..fb129db3dda95ff55f794e2deaa66b9d210dfd6f 100644
--- a/theories/base.v
+++ b/theories/base.v
@@ -94,6 +94,9 @@ Proof. split; repeat intro; congruence. Qed.
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
Class Equiv A := equiv: relation A.
+(* No Hint Mode set because of Coq bug #5735
+Hint Mode Equiv ! : typeclass_instances. *)
+
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
Notation "( X ≡)" := (equiv X) (only parsing) : C_scope.
@@ -108,10 +111,12 @@ with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x ≡ y → x = y.
+Hint Mode LeibnizEquiv ! - : typeclass_instances.
+
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv A _)} (x y : A) :
x ≡ y ↔ x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
-
+
Ltac fold_leibniz := repeat
match goal with
| H : context [ @equiv ?A _ _ _ ] |- _ =>
@@ -149,12 +154,14 @@ propositions. For example to declare a parameter expressing decidable equality
on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing
[decide (x = y)]. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
+Hint Mode Decision ! : typeclass_instances.
Arguments decide _ {_} : assert.
Notation EqDecision A := (∀ x y : A, Decision (x = y)).
(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
+Hint Mode Inhabited ! : typeclass_instances.
Arguments populate {_} _ : assert.
(** ** Proof irrelevant types *)
@@ -162,6 +169,7 @@ Arguments populate {_} _ : assert.
elements of the type are equal. We use this notion only used for propositions,
but by universe polymorphism we can generalize it. *)
Class ProofIrrel (A : Type) : Prop := proof_irrel (x y : A) : x = y.
+Hint Mode ProofIrrel ! : typeclass_instances.
(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
@@ -625,14 +633,17 @@ relations on collections: the empty collection [∅], the union [(∪)],
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Class Empty A := empty: A.
+Hint Mode Empty ! : typeclass_instances.
Notation "∅" := empty : C_scope.
Instance empty_inhabited `(Empty A) : Inhabited A := populate ∅.
Class Top A := top : A.
+Hint Mode Top ! : typeclass_instances.
Notation "⊤" := top : C_scope.
Class Union A := union: A → A → A.
+Hint Mode Union ! : typeclass_instances.
Instance: Params (@union) 2.
Infix "∪" := union (at level 50, left associativity) : C_scope.
Notation "(∪)" := union (only parsing) : C_scope.
@@ -650,6 +661,7 @@ Arguments union_list _ _ _ !_ / : assert.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃ l") : C_scope.
Class Intersection A := intersection: A → A → A.
+Hint Mode Intersection ! : typeclass_instances.
Instance: Params (@intersection) 2.
Infix "∩" := intersection (at level 40) : C_scope.
Notation "(∩)" := intersection (only parsing) : C_scope.
@@ -657,6 +669,7 @@ Notation "( x ∩)" := (intersection x) (only parsing) : C_scope.
Notation "(∩ x )" := (λ y, intersection y x) (only parsing) : C_scope.
Class Difference A := difference: A → A → A.
+Hint Mode Difference ! : typeclass_instances.
Instance: Params (@difference) 2.
Infix "∖" := difference (at level 40, left associativity) : C_scope.
Notation "(∖)" := difference (only parsing) : C_scope.
@@ -670,6 +683,7 @@ Infix "∖*∖**" := (zip_with (prod_zip (∖) (∖*)))
(at level 50, left associativity) : C_scope.
Class Singleton A B := singleton: A → B.
+Hint Mode Singleton - ! : typeclass_instances.
Instance: Params (@singleton) 3.
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
Notation "{[ x ; y ; .. ; z ]}" :=
@@ -681,6 +695,7 @@ Notation "{[ x , y , z ]}" := (singleton (x,y,z))
(at level 1, y at next level, z at next level) : C_scope.
Class SubsetEq A := subseteq: relation A.
+Hint Mode SubsetEq ! : typeclass_instances.
Instance: Params (@subseteq) 2.
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
@@ -720,8 +735,10 @@ Notation "X ⊂ Y ⊂ Z" := (X ⊂ Y ∧ Y ⊂ Z) (at level 70, Y at next level)
is used to create finite maps, finite sets, etc, and is typically different from
the order [(⊆)]. *)
Class Lexico A := lexico: relation A.
+Hint Mode Lexico ! : typeclass_instances.
Class ElemOf A B := elem_of: A → B → Prop.
+Hint Mode ElemOf - ! : typeclass_instances.
Instance: Params (@elem_of) 3.
Infix "∈" := elem_of (at level 70) : C_scope.
Notation "(∈)" := elem_of (only parsing) : C_scope.
@@ -733,6 +750,7 @@ Notation "( x ∉)" := (λ X, x ∉ X) (only parsing) : C_scope.
Notation "(∉ X )" := (λ x, x ∉ X) (only parsing) : C_scope.
Class Disjoint A := disjoint : A → A → Prop.
+Hint Mode Disjoint ! : typeclass_instances.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
@@ -749,6 +767,7 @@ Hint Extern 0 (_ ⊥ _) => symmetry; eassumption.
Hint Extern 0 (_ ⊥* _) => symmetry; eassumption.
Class DisjointE E A := disjointE : E → A → A → Prop.
+Hint Mode DisjointE - ! : typeclass_instances.
Instance: Params (@disjointE) 4.
Notation "X ⊥{ Γ } Y" := (disjointE Γ X Y)
(at level 70, format "X ⊥{ Γ } Y") : C_scope.
@@ -765,11 +784,14 @@ Notation "Xs ⊥{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
Hint Extern 0 (_ ⊥{_} _) => symmetry; eassumption.
Class DisjointList A := disjoint_list : list A → Prop.
+Hint Mode DisjointList ! : typeclass_instances.
Instance: Params (@disjoint_list) 2.
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥ Xs") : C_scope.
Section disjoint_list.
Context `{Disjoint A, Union A, Empty A}.
+ Implicit Types X : A.
+
Inductive disjoint_list_default : DisjointList A :=
| disjoint_nil_2 : ⊥ (@nil A)
| disjoint_cons_2 (X : A) (Xs : list A) : X ⊥ ⋃ Xs → ⊥ Xs → ⊥ (X :: Xs).
@@ -782,8 +804,10 @@ Section disjoint_list.
End disjoint_list.
Class Filter A B := filter: ∀ (P : A → Prop) `{∀ x, Decision (P x)}, B → B.
+Hint Mode Filter - ! : typeclass_instances.
Class UpClose A B := up_close : A → B.
+Hint Mode UpClose - ! : typeclass_instances.
Notation "↑ x" := (up_close x) (at level 20, format "↑ x").
(** * Monadic operations *)
@@ -850,6 +874,7 @@ Notation "'guard' P 'as' H ; o" := (mguard P (λ H, o))
on maps. In the file [fin_maps] we will axiomatize finite maps.
The function look up [m !! k] should yield the element at key [k] in [m]. *)
Class Lookup (K A M : Type) := lookup: K → M → option A.
+Hint Mode Lookup - - ! : typeclass_instances.
Instance: Params (@lookup) 4.
Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
@@ -859,12 +884,14 @@ Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch, assert.
(** The singleton map *)
Class SingletonM K A M := singletonM: K → A → M.
+Hint Mode SingletonM - - ! : typeclass_instances.
Instance: Params (@singletonM) 5.
Notation "{[ k := a ]}" := (singletonM k a) (at level 1) : C_scope.
(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
Class Insert (K A M : Type) := insert: K → A → M → M.
+Hint Mode Insert - - ! : typeclass_instances.
Instance: Params (@insert) 5.
Notation "<[ k := a ]>" := (insert k a)
(at level 5, right associativity, format "<[ k := a ]>") : C_scope.
@@ -874,12 +901,14 @@ Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch, assert.
[m]. If the key [k] is not a member of [m], the original map should be
returned. *)
Class Delete (K M : Type) := delete: K → M → M.
+Hint Mode Delete - ! : typeclass_instances.
Instance: Params (@delete) 4.
Arguments delete _ _ _ !_ !_ / : simpl nomatch, assert.
(** The function [alter f k m] should update the value at key [k] using the
function [f], which is called with the original value. *)
Class Alter (K A M : Type) := alter: (A → A) → K → M → M.
+Hint Mode Alter - - ! : typeclass_instances.
Instance: Params (@alter) 5.
Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch, assert.
@@ -889,12 +918,14 @@ if [k] is not a member of [m]. The value at [k] should be deleted if [f]
yields [None]. *)
Class PartialAlter (K A M : Type) :=
partial_alter: (option A → option A) → K → M → M.
+Hint Mode PartialAlter - - ! : typeclass_instances.
Instance: Params (@partial_alter) 4.
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch, assert.
(** The function [dom C m] should yield the domain of [m]. That is a finite
collection of type [C] that contains the keys that are a member of [m]. *)
Class Dom (M C : Type) := dom: M → C.
+Hint Mode Dom ! ! : typeclass_instances.
Instance: Params (@dom) 3.
Arguments dom : clear implicits.
Arguments dom {_} _ {_} !_ / : simpl nomatch, assert.
@@ -903,6 +934,7 @@ Arguments dom {_} _ {_} !_ / : simpl nomatch, assert.
constructing a new map whose value at key [k] is [f (m1 !! k) (m2 !! k)].*)
Class Merge (M : Type → Type) :=
merge: ∀ {A B C}, (option A → option B → option C) → M A → M B → M C.
+Hint Mode Merge ! : typeclass_instances.
Instance: Params (@merge) 4.
Arguments merge _ _ _ _ _ _ !_ !_ / : simpl nomatch, assert.
@@ -911,17 +943,20 @@ and [m2] using the function [f] to combine values of members that are in
both [m1] and [m2]. *)
Class UnionWith (A M : Type) :=
union_with: (A → A → option A) → M → M → M.
+Hint Mode UnionWith - ! : typeclass_instances.
Instance: Params (@union_with) 3.
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
intersection_with: (A → A → option A) → M → M → M.
+Hint Mode IntersectionWith - ! : typeclass_instances.
Instance: Params (@intersection_with) 3.
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.
Class DifferenceWith (A M : Type) :=
difference_with: (A → A → option A) → M → M → M.
+Hint Mode DifferenceWith - ! : typeclass_instances.
Instance: Params (@difference_with) 3.
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.
@@ -930,6 +965,7 @@ Definition intersection_with_list `{IntersectionWith A M}
Arguments intersection_with_list _ _ _ _ _ !_ / : assert.
Class LookupE (E K A M : Type) := lookupE: E → K → M → option A.
+Hint Mode LookupE - - - ! : typeclass_instances.
Instance: Params (@lookupE) 6.
Notation "m !!{ Γ } i" := (lookupE Γ i m)
(at level 20, format "m !!{ Γ } i") : C_scope.
@@ -937,6 +973,7 @@ Notation "(!!{ Γ } )" := (lookupE Γ) (only parsing, Γ at level 1) : C_scope.
Arguments lookupE _ _ _ _ _ _ !_ !_ / : simpl nomatch, assert.
Class InsertE (E K A M : Type) := insertE: E → K → A → M → M.
+Hint Mode InsertE - - - ! : typeclass_instances.
Instance: Params (@insertE) 6.
Notation "<[ k := a ]{ Γ }>" := (insertE Γ k a)
(at level 5, right associativity, format "<[ k := a ]{ Γ }>") : C_scope.
@@ -963,6 +1000,7 @@ Class Collection A C `{ElemOf A C, Empty C, Singleton A C,
enumerated as a list. These elements, given by the [elements] function, may be
in any order and should not contain duplicates. *)
Class Elements A C := elements: C → list A.
+Hint Mode Elements - ! : typeclass_instances.
Instance: Params (@elements) 3.
(** We redefine the standard library's [In] and [NoDup] using type classes. *)
@@ -998,6 +1036,7 @@ Class FinCollection A C `{ElemOf A C, Empty C, Singleton A C, Union C,
NoDup_elements X : NoDup (elements X)
}.
Class Size C := size: C → nat.
+Hint Mode Size ! : typeclass_instances.
Arguments size {_ _} !_ / : simpl nomatch, assert.
Instance: Params (@size) 2.
@@ -1025,6 +1064,7 @@ Class CollectionMonad M `{∀ A, ElemOf A (M A),
will later prove that [fresh] is [Proper] with respect to the induced setoid
equality on collections. *)
Class Fresh A C := fresh: C → A.
+Hint Mode Fresh - ! : typeclass_instances.
Instance: Params (@fresh) 3.
Class FreshSpec A C `{ElemOf A C,
Empty C, Singleton A C, Union C, Fresh A C} : Prop := {
@@ -1035,5 +1075,6 @@ Class FreshSpec A C `{ElemOf A C,
(** * Miscellaneous *)
Class Half A := half: A → A.
+Hint Mode Half ! : typeclass_instances.
Notation "½" := half : C_scope.
Notation "½*" := (fmap (M:=list) half) : C_scope.
diff --git a/theories/coPset.v b/theories/coPset.v
index 68b8466fea08fbbc555e05fac3fd95339e275e7e..4af9f0233e92a20b4d420be0b68ec9d769b87a0d 100644
--- a/theories/coPset.v
+++ b/theories/coPset.v
@@ -427,7 +427,7 @@ Proof.
rewrite !coPset_finite_spec; destruct X as [t Ht]; simpl; clear Ht.
induction t as [[]|]; simpl; rewrite ?coPset_finite_node, ?andb_True; tauto.
Qed.
-Lemma coPset_split X :
+Lemma coPset_split (X : coPset) :
¬set_finite X →
∃ X1 X2, X = X1 ∪ X2 ∧ X1 ∩ X2 = ∅ ∧ ¬set_finite X1 ∧ ¬set_finite X2.
Proof.
diff --git a/theories/collections.v b/theories/collections.v
index bb6c4b5fcaff5d76ffc8deb361b0922ac5b6a773..134af648961b502059787fe706048e0cb79dcfd7 100644
--- a/theories/collections.v
+++ b/theories/collections.v
@@ -145,9 +145,9 @@ Section set_unfold_simple.
Implicit Types x y : A.
Implicit Types X Y : C.
- Global Instance set_unfold_empty x : SetUnfold (x ∈ ∅) False.
+ Global Instance set_unfold_empty x : SetUnfold (x ∈ (∅ : C)) False.
Proof. constructor. split. apply not_elem_of_empty. done. Qed.
- Global Instance set_unfold_singleton x y : SetUnfold (x ∈ {[ y ]}) (x = y).
+ Global Instance set_unfold_singleton x y : SetUnfold (x ∈ ({[ y ]} : C)) (x = y).
Proof. constructor; apply elem_of_singleton. Qed.
Global Instance set_unfold_union x X Y P Q :
SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ∪ Y) (P ∨ Q).
@@ -161,30 +161,30 @@ Section set_unfold_simple.
(∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (∅ ≡ X) (∀ x, ¬P x) | 5.
Proof.
intros ?; constructor. unfold equiv, collection_equiv.
- pose proof not_elem_of_empty; naive_solver.
+ pose proof (not_elem_of_empty (C:=C)); naive_solver.
Qed.
- Global Instance set_unfold_equiv_empty_r (P : A → Prop) :
+ Global Instance set_unfold_equiv_empty_r (P : A → Prop) X :
(∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (X ≡ ∅) (∀ x, ¬P x) | 5.
Proof.
intros ?; constructor. unfold equiv, collection_equiv.
- pose proof not_elem_of_empty; naive_solver.
+ pose proof (not_elem_of_empty (C:=C)); naive_solver.
Qed.
- Global Instance set_unfold_equiv (P Q : A → Prop) :
+ Global Instance set_unfold_equiv (P Q : A → Prop) X :
(∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
SetUnfold (X ≡ Y) (∀ x, P x ↔ Q x) | 10.
Proof. constructor. apply forall_proper; naive_solver. Qed.
- Global Instance set_unfold_subseteq (P Q : A → Prop) :
+ Global Instance set_unfold_subseteq (P Q : A → Prop) X Y :
(∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
SetUnfold (X ⊆ Y) (∀ x, P x → Q x).
Proof. constructor. apply forall_proper; naive_solver. Qed.
- Global Instance set_unfold_subset (P Q : A → Prop) :
+ Global Instance set_unfold_subset (P Q : A → Prop) X :
(∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
SetUnfold (X ⊂ Y) ((∀ x, P x → Q x) ∧ ¬∀ x, Q x → P x).
Proof.
constructor. unfold strict.
repeat f_equiv; apply forall_proper; naive_solver.
Qed.
- Global Instance set_unfold_disjoint (P Q : A → Prop) :
+ Global Instance set_unfold_disjoint (P Q : A → Prop) X Y :
(∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
SetUnfold (X ⊥ Y) (∀ x, P x → Q x → False).
Proof. constructor. unfold disjoint, collection_disjoint. naive_solver. Qed.
@@ -195,10 +195,10 @@ Section set_unfold_simple.
Global Instance set_unfold_equiv_empty_l_L X (P : A → Prop) :
(∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (∅ = X) (∀ x, ¬P x) | 5.
Proof. constructor. unfold_leibniz. by apply set_unfold_equiv_empty_l. Qed.
- Global Instance set_unfold_equiv_empty_r_L (P : A → Prop) :
+ Global Instance set_unfold_equiv_empty_r_L (P : A → Prop) X :
(∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (X = ∅) (∀ x, ¬P x) | 5.
Proof. constructor. unfold_leibniz. by apply set_unfold_equiv_empty_r. Qed.
- Global Instance set_unfold_equiv_L (P Q : A → Prop) :
+ Global Instance set_unfold_equiv_L (P Q : A → Prop) X Y :
(∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
SetUnfold (X = Y) (∀ x, P x ↔ Q x) | 10.
Proof. constructor. unfold_leibniz. by apply set_unfold_equiv. Qed.
@@ -224,20 +224,20 @@ Section set_unfold.
End set_unfold.
Section set_unfold_monad.
- Context `{CollectionMonad M} {A : Type}.
- Implicit Types x y : A.
+ Context `{CollectionMonad M}.
- Global Instance set_unfold_ret x y : SetUnfold (x ∈ mret y) (x = y).
+ Global Instance set_unfold_ret {A} (x y : A) :
+ SetUnfold (x ∈ mret (M:=M) y) (x = y).
Proof. constructor; apply elem_of_ret. Qed.
- Global Instance set_unfold_bind {B} (f : A → M B) X (P Q : A → Prop) :
+ Global Instance set_unfold_bind {A B} (f : A → M B) X (P Q : A → Prop) :
(∀ y, SetUnfold (y ∈ X) (P y)) → (∀ y, SetUnfold (x ∈ f y) (Q y)) →
SetUnfold (x ∈ X ≫= f) (∃ y, Q y ∧ P y).
Proof. constructor. rewrite elem_of_bind; naive_solver. Qed.
- Global Instance set_unfold_fmap {B} (f : A → B) X (P : A → Prop) :
+ Global Instance set_unfold_fmap {A B} (f : A → B) (X : M A) (P : A → Prop) :
(∀ y, SetUnfold (y ∈ X) (P y)) →
SetUnfold (x ∈ f <$> X) (∃ y, x = f y ∧ P y).
Proof. constructor. rewrite elem_of_fmap; naive_solver. Qed.
- Global Instance set_unfold_join (X : M (M A)) (P : M A → Prop) :
+ Global Instance set_unfold_join {A} (X : M (M A)) (P : M A → Prop) :
(∀ Y, SetUnfold (Y ∈ X) (P Y)) → SetUnfold (x ∈ mjoin X) (∃ Y, x ∈ Y ∧ P Y).
Proof. constructor. rewrite elem_of_join; naive_solver. Qed.
End set_unfold_monad.
@@ -381,7 +381,7 @@ Section simple_collection.
Proof. set_solver. Qed.
Lemma elem_of_equiv_empty X : X ≡ ∅ ↔ ∀ x, x ∉ X.
Proof. set_solver. Qed.
- Lemma elem_of_empty x : x ∈ ∅ ↔ False.
+ Lemma elem_of_empty x : x ∈ (∅ : C) ↔ False.
Proof. set_solver. Qed.
Lemma equiv_empty X : X ⊆ ∅ → X ≡ ∅.
Proof. set_solver. Qed.
@@ -393,15 +393,15 @@ Section simple_collection.
Proof. set_solver. Qed.
(** Singleton *)
- Lemma elem_of_singleton_1 x y : x ∈ {[y]} → x = y.
+ Lemma elem_of_singleton_1 x y : x ∈ ({[y]} : C) → x = y.
Proof. by rewrite elem_of_singleton. Qed.
- Lemma elem_of_singleton_2 x y : x = y → x ∈ {[y]}.
+ Lemma elem_of_singleton_2 x y : x = y → x ∈ ({[y]} : C).
Proof. by rewrite elem_of_singleton. Qed.
Lemma elem_of_subseteq_singleton x X : x ∈ X ↔ {[ x ]} ⊆ X.
Proof. set_solver. Qed.
Lemma non_empty_singleton x : ({[ x ]} : C) ≢ ∅.
Proof. set_solver. Qed.
- Lemma not_elem_of_singleton x y : x ∉ {[ y ]} ↔ x ≠y.
+ Lemma not_elem_of_singleton x y : x ∉ ({[ y ]} : C) ↔ x ≠y.
Proof. by rewrite elem_of_singleton. Qed.
(** Disjointness *)
@@ -512,7 +512,7 @@ Section simple_collection.
Proof. unfold_leibniz. apply non_empty_inhabited. Qed.
(** Singleton *)
- Lemma non_empty_singleton_L x : {[ x ]} ≠∅.
+ Lemma non_empty_singleton_L x : {[ x ]} ≠(∅ : C).
Proof. unfold_leibniz. apply non_empty_singleton. Qed.
(** Big unions *)
@@ -554,6 +554,7 @@ End simple_collection.
(** * Collections with [∪], [∩], [∖], [∅] and [{[_]}] *)
Section collection.
Context `{Collection A C}.
+ Implicit Types x y : A.
Implicit Types X Y : C.
(** Intersection *)
@@ -654,7 +655,7 @@ Section collection.
Global Instance intersection_empty_r_L: RightAbsorb ((=) : relation C) ∅ (∩).
Proof. intros ?. unfold_leibniz. apply (right_absorb _ _). Qed.
- Lemma intersection_singletons_L x : {[x]} ∩ {[x]} = {[x]}.
+ Lemma intersection_singletons_L x : {[x]} ∩ {[x]} = ({[x]} : C).
Proof. unfold_leibniz. apply intersection_singletons. Qed.
Lemma union_intersection_l_L X Y Z : X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z).
@@ -719,7 +720,7 @@ Section collection.
{[x]} ∪ (X ∖ Y) ≡ ({[x]} ∪ X) ∖ (Y ∖ {[x]}).
Proof.
intro y; split; intros Hy; [ set_solver | ].
- destruct (decide (y ∈ {[x]})); set_solver.
+ destruct (decide (y ∈ ({[x]} : C))); set_solver.
Qed.
Context `{!LeibnizEquiv C}.
@@ -736,7 +737,6 @@ Section collection.
Lemma singleton_union_difference_L X Y x :
{[x]} ∪ (X ∖ Y) = ({[x]} ∪ X) ∖ (Y ∖ {[x]}).
Proof. unfold_leibniz. apply singleton_union_difference. Qed.
-
End dec.
End collection.
@@ -751,26 +751,26 @@ Section of_option_list.
Context `{SimpleCollection A C}.
Implicit Types l : list A.
- Lemma elem_of_of_option (x : A) mx: x ∈ of_option mx ↔ mx = Some x.
+ Lemma elem_of_of_option (x : A) mx: x ∈ of_option (C:=C) mx ↔ mx = Some x.
Proof. destruct mx; set_solver. Qed.
- Lemma not_elem_of_of_option (x : A) mx: x ∉ of_option mx ↔ mx ≠Some x.
+ Lemma not_elem_of_of_option (x : A) mx: x ∉ of_option (C:=C) mx ↔ mx ≠Some x.
Proof. by rewrite elem_of_of_option. Qed.
- Lemma elem_of_of_list (x : A) l : x ∈ of_list l ↔ x ∈ l.
+ Lemma elem_of_of_list (x : A) l : x ∈ of_list (C:=C) l ↔ x ∈ l.
Proof.
split.
- induction l; simpl; [by rewrite elem_of_empty|].
rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto.
- induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
Qed.
- Lemma not_elem_of_of_list (x : A) l : x ∉ of_list l ↔ x ∉ l.
+ Lemma not_elem_of_of_list (x : A) l : x ∉ of_list (C:=C) l ↔ x ∉ l.
Proof. by rewrite elem_of_of_list. Qed.
Global Instance set_unfold_of_option (mx : option A) x :
- SetUnfold (x ∈ of_option mx) (mx = Some x).
+ SetUnfold (x ∈ of_option (C:=C) mx) (mx = Some x).
Proof. constructor; apply elem_of_of_option. Qed.
Global Instance set_unfold_of_list (l : list A) x P :
- SetUnfold (x ∈ l) P → SetUnfold (x ∈ of_list l) P.
+ SetUnfold (x ∈ l) P → SetUnfold (x ∈ of_list (C:=C) l) P.
Proof. constructor. by rewrite elem_of_of_list, (set_unfold (x ∈ l) P). Qed.
Lemma of_list_nil : of_list (C:=C) [] = ∅.
@@ -810,7 +810,7 @@ Section collection_monad_base.
rewrite !elem_of_equiv_empty; setoid_rewrite elem_of_guard.
destruct (decide P); naive_solver.
Qed.
- Global Instance set_unfold_guard `{Decision P} {A} (x : A) X Q :
+ Global Instance set_unfold_guard `{Decision P} {A} (x : A) (X : M A) Q :
SetUnfold (x ∈ X) Q → SetUnfold (x ∈ guard P; X) (P ∧ Q).
Proof. constructor. by rewrite elem_of_guard, (set_unfold (x ∈ X) Q). Qed.
Lemma bind_empty {A B} (f : A → M B) X :
@@ -824,11 +824,12 @@ Definition set_Forall `{ElemOf A C} (P : A → Prop) (X : C) := ∀ x, x ∈ X
Definition set_Exists `{ElemOf A C} (P : A → Prop) (X : C) := ∃ x, x ∈ X ∧ P x.
Section quantifiers.
- Context `{SimpleCollection A B} (P : A → Prop).
+ Context `{SimpleCollection A C} (P : A → Prop).
+ Implicit Types X Y : C.
- Lemma set_Forall_empty : set_Forall P ∅.
+ Lemma set_Forall_empty : set_Forall P (∅ : C).
Proof. unfold set_Forall. set_solver. Qed.
- Lemma set_Forall_singleton x : set_Forall P {[ x ]} ↔ P x.
+ Lemma set_Forall_singleton x : set_Forall P ({[ x ]} : C) ↔ P x.
Proof. unfold set_Forall. set_solver. Qed.
Lemma set_Forall_union X Y :
set_Forall P X → set_Forall P Y → set_Forall P (X ∪ Y).
@@ -838,9 +839,9 @@ Section quantifiers.
Lemma set_Forall_union_inv_2 X Y : set_Forall P (X ∪ Y) → set_Forall P Y.
Proof. unfold set_Forall. set_solver. Qed.
- Lemma set_Exists_empty : ¬set_Exists P ∅.
+ Lemma set_Exists_empty : ¬set_Exists P (∅ : C).
Proof. unfold set_Exists. set_solver. Qed.
- Lemma set_Exists_singleton x : set_Exists P {[ x ]} ↔ P x.
+ Lemma set_Exists_singleton x : set_Exists P ({[ x ]} : C) ↔ P x.
Proof. unfold set_Exists. set_solver. Qed.
Lemma set_Exists_union_1 X Y : set_Exists P X → set_Exists P (X ∪ Y).
Proof. unfold set_Exists. set_solver. Qed.
@@ -852,7 +853,8 @@ Section quantifiers.
End quantifiers.
Section more_quantifiers.
- Context `{SimpleCollection A B}.
+ Context `{SimpleCollection A C}.
+ Implicit Types X : C.
Lemma set_Forall_impl (P Q : A → Prop) X :
set_Forall P X → (∀ x, P x → Q x) → set_Forall Q X.
@@ -987,15 +989,17 @@ End collection_monad.
Definition set_finite `{ElemOf A B} (X : B) := ∃ l : list A, ∀ x, x ∈ X → x ∈ l.
Section finite.
- Context `{SimpleCollection A B}.
+ Context `{SimpleCollection A C}.
+ Implicit Types X Y : C.
+
Global Instance set_finite_subseteq :
- Proper (flip (⊆) ==> impl) (@set_finite A B _).
+ Proper (flip (⊆) ==> impl) (@set_finite A C _).
Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed.
- Global Instance set_finite_proper : Proper ((≡) ==> iff) (@set_finite A B _).
+ Global Instance set_finite_proper : Proper ((≡) ==> iff) (@set_finite A C _).
Proof. intros X Y HX; apply exist_proper. by setoid_rewrite HX. Qed.
- Lemma empty_finite : set_finite ∅.
+ Lemma empty_finite : set_finite (∅ : C).
Proof. by exists []; intros ?; rewrite elem_of_empty. Qed.
- Lemma singleton_finite (x : A) : set_finite {[ x ]}.
+ Lemma singleton_finite (x : A) : set_finite ({[ x ]} : C).
Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed.
Lemma union_finite X Y : set_finite X → set_finite Y → set_finite (X ∪ Y).
Proof.
@@ -1009,7 +1013,9 @@ Section finite.
End finite.
Section more_finite.
- Context `{Collection A B}.
+ Context `{Collection A C}.
+ Implicit Types X Y : C.
+
Lemma intersection_finite_l X Y : set_finite X → set_finite (X ∩ Y).
Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed.
Lemma intersection_finite_r X Y : set_finite Y → set_finite (X ∩ Y).
@@ -1038,14 +1044,15 @@ Section seq_set.
Implicit Types start len x : nat.
Lemma elem_of_seq_set start len x :
- x ∈ seq_set start len ↔ start ≤ x < start + len.
+ x ∈ seq_set (C:=C) start len ↔ start ≤ x < start + len.
Proof.
revert start. induction len as [|len IH]; intros start; simpl.
- rewrite elem_of_empty. omega.
- rewrite elem_of_union, elem_of_singleton, IH. omega.
Qed.
- Lemma seq_set_S_disjoint start len : {[ start + len ]} ⊥ seq_set start len.
+ Lemma seq_set_S_disjoint start len :
+ {[ start + len ]} ⊥ seq_set (C:=C) start len.
Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. omega. Qed.
Lemma seq_set_S_union start len :
@@ -1055,7 +1062,7 @@ Section seq_set.
Qed.
Lemma seq_set_S_union_L `{!LeibnizEquiv C} start len :
- seq_set start (S len) = {[ start + len ]} ∪ seq_set start len.
+ seq_set start (C:=C) (S len) = {[ start + len ]} ∪ seq_set start len.
Proof. unfold_leibniz. apply seq_set_S_union. Qed.
End seq_set.
@@ -1067,6 +1074,7 @@ Typeclasses Opaque minimal.
Section minimal.
Context `{SimpleCollection A C} {R : relation A}.
+ Implicit Types X Y : C.
Global Instance minimal_proper x : Proper (@equiv C _ ==> iff) (minimal R x).
Proof. intros X X' y; unfold minimal; set_solver. Qed.
@@ -1085,11 +1093,11 @@ Section minimal.
minimal R x X ↔ ∀ y, y ∈ X → ¬R y x.
Proof. unfold minimal; split; [eauto using minimal_strict_1|naive_solver]. Qed.
- Lemma empty_minimal x : minimal R x ∅.
+ Lemma empty_minimal x : minimal R x (∅ : C).
Proof. unfold minimal; set_solver. Qed.
- Lemma singleton_minimal x : minimal R x {[ x ]}.
+ Lemma singleton_minimal x : minimal R x ({[ x ]} : C).
Proof. unfold minimal; set_solver. Qed.
- Lemma singleton_minimal_not_above y x : ¬R y x → minimal R x {[ y ]}.
+ Lemma singleton_minimal_not_above y x : ¬R y x → minimal R x ({[ y ]} : C).
Proof. unfold minimal; set_solver. Qed.
Lemma union_minimal X Y x :
minimal R x X → minimal R x Y → minimal R x (X ∪ Y).
diff --git a/theories/fin_collections.v b/theories/fin_collections.v
index f5061160adb40a3743f397add31630fca38578ea..ddb1220c05c4c63ee3f8eab8b0905dfcc5c7a040 100644
--- a/theories/fin_collections.v
+++ b/theories/fin_collections.v
@@ -63,14 +63,14 @@ Proof.
intros y; rewrite elem_of_elements, elem_of_union, elem_of_singleton.
by rewrite elem_of_cons, elem_of_elements.
Qed.
-Lemma elements_singleton x : elements {[ x ]} = [x].
+Lemma elements_singleton x : elements ({[ x ]} : C) = [x].
Proof.
apply Permutation_singleton. by rewrite <-(right_id ∅ (∪) {[x]}),
elements_union_singleton, elements_empty by set_solver.
Qed.
Lemma elements_submseteq X Y : X ⊆ Y → elements X ⊆+ elements Y.
Proof.
- intros; apply NoDup_submseteq; auto using NoDup_elements.
+ intros; apply NoDup_submseteq; eauto using NoDup_elements.
intros x. rewrite !elem_of_elements; auto.
Qed.
@@ -106,7 +106,7 @@ Proof.
contradict Hsz. rewrite HX, size_empty; lia.
Qed.
-Lemma size_singleton (x : A) : size {[ x ]} = 1.
+Lemma size_singleton (x : A) : size ({[ x ]} : C) = 1.
Proof. unfold size, collection_size. simpl. by rewrite elements_singleton. Qed.
Lemma size_singleton_inv X x y : size X = 1 → x ∈ X → y ∈ X → x = y.
Proof.
@@ -200,9 +200,9 @@ Proof.
{ destruct IH as (x' & Hx' & Hmin); [set_solver|].
destruct (decide (R x x')).
- exists x; split; [set_solver|].
- eauto using union_minimal, singleton_minimal, minimal_weaken.
+ eauto using (union_minimal (C:=C)), (singleton_minimal (C:=C)), minimal_weaken.
- exists x'; split; [set_solver|].
- auto using union_minimal, singleton_minimal_not_above. }
+ eauto using (union_minimal (C:=C)), (singleton_minimal_not_above (C:=C)). }
exists x; split; [set_solver|].
rewrite HX, (right_id _ (∪)). apply singleton_minimal.
Qed.
diff --git a/theories/fin_map_dom.v b/theories/fin_map_dom.v
index 5be84036978123a40071174129d392b25eda7022..68632ad6c64daad09798be1d4cd5545e9eff520c 100644
--- a/theories/fin_map_dom.v
+++ b/theories/fin_map_dom.v
@@ -61,7 +61,7 @@ Proof. rewrite (dom_insert _). set_solver. Qed.
Lemma dom_insert_subseteq_compat_l {A} (m : M A) i x X :
X ⊆ dom D m → X ⊆ dom D (<[i:=x]>m).
Proof. intros. trans (dom D m); eauto using dom_insert_subseteq. Qed.
-Lemma dom_singleton {A} (i : K) (x : A) : dom D {[i := x]} ≡ {[ i ]}.
+Lemma dom_singleton {A} (i : K) (x : A) : dom D ({[i := x]} : M A) ≡ {[ i ]}.
Proof. rewrite <-insert_empty, dom_insert, dom_empty; set_solver. Qed.
Lemma dom_delete {A} (m : M A) i : dom D (delete i m) ≡ dom D m ∖ {[ i ]}.
Proof.
@@ -100,7 +100,7 @@ Proof.
unfold is_Some. setoid_rewrite lookup_difference_Some.
destruct (m2 !! i); naive_solver.
Qed.
-Lemma dom_fmap {A B} (f : A → B) m : dom D (f <$> m) ≡ dom D m.
+Lemma dom_fmap {A B} (f : A → B) (m : M A) : dom D (f <$> m) ≡ dom D m.
Proof.
apply elem_of_equiv. intros i.
rewrite !elem_of_dom, lookup_fmap, <-!not_eq_None_Some.
@@ -109,7 +109,8 @@ Qed.
Lemma dom_finite {A} (m : M A) : set_finite (dom D m).
Proof.
induction m using map_ind; rewrite ?dom_empty, ?dom_insert;
- eauto using empty_finite, union_finite, singleton_finite.
+ eauto using (empty_finite (C:=D)), (union_finite (C:=D)),
+ (singleton_finite (C:=D)).
Qed.
Context `{!LeibnizEquiv D}.
@@ -121,7 +122,7 @@ Lemma dom_alter_L {A} f (m : M A) i : dom D (alter f i m) = dom D m.
Proof. unfold_leibniz; apply dom_alter. Qed.
Lemma dom_insert_L {A} (m : M A) i x : dom D (<[i:=x]>m) = {[ i ]} ∪ dom D m.
Proof. unfold_leibniz; apply dom_insert. Qed.
-Lemma dom_singleton_L {A} (i : K) (x : A) : dom D {[i := x]} = {[ i ]}.
+Lemma dom_singleton_L {A} (i : K) (x : A) : dom D ({[i := x]} : M A) = {[ i ]}.
Proof. unfold_leibniz; apply dom_singleton. Qed.
Lemma dom_delete_L {A} (m : M A) i : dom D (delete i m) = dom D m ∖ {[ i ]}.
Proof. unfold_leibniz; apply dom_delete. Qed.
@@ -132,6 +133,6 @@ Lemma dom_intersection_L {A} (m1 m2 : M A) :
Proof. unfold_leibniz; apply dom_intersection. Qed.
Lemma dom_difference_L {A} (m1 m2 : M A) : dom D (m1 ∖ m2) = dom D m1 ∖ dom D m2.
Proof. unfold_leibniz; apply dom_difference. Qed.
-Lemma dom_fmap_L {A B} (f : A → B) m : dom D (f <$> m) = dom D m.
+Lemma dom_fmap_L {A B} (f : A → B) (m : M A) : dom D (f <$> m) = dom D m.
Proof. unfold_leibniz; apply dom_fmap. Qed.
End fin_map_dom.
diff --git a/theories/fin_maps.v b/theories/fin_maps.v
index e9607aeffd30e960c7e03a85c268882b44da9356..fcaba6ce290f1c215813c8dbad6536d1a3eb3b6c 100644
--- a/theories/fin_maps.v
+++ b/theories/fin_maps.v
@@ -26,6 +26,8 @@ which enables us to give a generic implementation of [union_with],
[intersection_with], and [difference_with]. *)
Class FinMapToList K A M := map_to_list: M → list (K * A).
+Hint Mode FinMapToList ! - - : typeclass_instances.
+Hint Mode FinMapToList - - ! : typeclass_instances.
Class FinMap K M `{FMap M, ∀ A, Lookup K A (M A), ∀ A, Empty (M A), ∀ A,
PartialAlter K A (M A), OMap M, Merge M, ∀ A, FinMapToList K A (M A),
@@ -213,7 +215,7 @@ Proof.
specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Global Instance map_included_preorder {A} (R : relation A) :
- PreOrder R → PreOrder (map_included R).
+ PreOrder R → PreOrder (map_included R : relation (M A)).
Proof.
split; [intros m i; by destruct (m !! i); simpl|].
intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
@@ -247,7 +249,7 @@ Lemma map_empty {A} (m : M A) : (∀ i, m !! i = None) → m = ∅.
Proof. intros Hm. apply map_eq. intros. by rewrite Hm, lookup_empty. Qed.
Lemma lookup_empty_is_Some {A} i : ¬is_Some ((∅ : M A) !! i).
Proof. rewrite lookup_empty. by inversion 1. Qed.
-Lemma lookup_empty_Some {A} i (x : A) : ¬∅ !! i = Some x.
+Lemma lookup_empty_Some {A} i (x : A) : ¬(∅ : M A) !! i = Some x.
Proof. by rewrite lookup_empty. Qed.
Lemma map_subset_empty {A} (m : M A) : m ⊄ ∅.
Proof.
@@ -319,9 +321,10 @@ Proof.
Qed.
(** ** Properties of the [alter] operation *)
-Lemma lookup_alter {A} (f : A → A) m i : alter f i m !! i = f <$> m !! i.
+Lemma lookup_alter {A} (f : A → A) (m : M A) i : alter f i m !! i = f <$> m !! i.
Proof. unfold alter. apply lookup_partial_alter. Qed.
-Lemma lookup_alter_ne {A} (f : A → A) m i j : i ≠j → alter f i m !! j = m !! j.
+Lemma lookup_alter_ne {A} (f : A → A) (m : M A) i j :
+ i ≠j → alter f i m !! j = m !! j.
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
Lemma alter_ext {A} (f g : A → A) (m : M A) i :
(∀ x, m !! i = Some x → f x = g x) → alter f i m = alter g i m.
@@ -335,7 +338,7 @@ Qed.
Lemma alter_commute {A} (f g : A → A) (m : M A) i j :
i ≠j → alter f i (alter g j m) = alter g j (alter f i m).
Proof. apply partial_alter_commute. Qed.
-Lemma lookup_alter_Some {A} (f : A → A) m i j y :
+Lemma lookup_alter_Some {A} (f : A → A) (m : M A) i j y :
alter f i m !! j = Some y ↔
(i = j ∧ ∃ x, m !! j = Some x ∧ y = f x) ∨ (i ≠j ∧ m !! j = Some y).
Proof.
@@ -343,16 +346,16 @@ Proof.
- rewrite lookup_alter. naive_solver (simplify_option_eq; eauto).
- rewrite lookup_alter_ne by done. naive_solver.
Qed.
-Lemma lookup_alter_None {A} (f : A → A) m i j :
+Lemma lookup_alter_None {A} (f : A → A) (m : M A) i j :
alter f i m !! j = None ↔ m !! j = None.
Proof.
by destruct (decide (i = j)) as [->|?];
rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
Qed.
-Lemma lookup_alter_is_Some {A} (f : A → A) m i j :
+Lemma lookup_alter_is_Some {A} (f : A → A) (m : M A) i j :
is_Some (alter f i m !! j) ↔ is_Some (m !! j).
Proof. by rewrite <-!not_eq_None_Some, lookup_alter_None. Qed.
-Lemma alter_id {A} (f : A → A) m i :
+Lemma alter_id {A} (f : A → A) (m : M A) i :
(∀ x, m !! i = Some x → f x = x) → alter f i m = m.
Proof.
intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
@@ -484,7 +487,7 @@ Proof.
- rewrite lookup_insert. destruct (m !! j); simpl; eauto.
- rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
Qed.
-Lemma insert_empty {A} i (x : A) : <[i:=x]>∅ = {[i := x]}.
+Lemma insert_empty {A} i (x : A) : <[i:=x]>(∅ : M A) = {[i := x]}.
Proof. done. Qed.
Lemma insert_non_empty {A} (m : M A) i x : <[i:=x]>m ≠∅.
Proof.
@@ -542,44 +545,47 @@ Qed.
(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
- {[i := x]} !! j = Some y ↔ i = j ∧ x = y.
+ ({[i := x]} : M A) !! j = Some y ↔ i = j ∧ x = y.
Proof.
rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
Qed.
-Lemma lookup_singleton_None {A} i j (x : A) : {[i := x]} !! j = None ↔ i ≠j.
+Lemma lookup_singleton_None {A} i j (x : A) :
+ ({[i := x]} : M A) !! j = None ↔ i ≠j.
Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed.
-Lemma lookup_singleton {A} i (x : A) : {[i := x]} !! i = Some x.
+Lemma lookup_singleton {A} i (x : A) : ({[i := x]} : M A) !! i = Some x.
Proof. by rewrite lookup_singleton_Some. Qed.
-Lemma lookup_singleton_ne {A} i j (x : A) : i ≠j → {[i := x]} !! j = None.
+Lemma lookup_singleton_ne {A} i j (x : A) :
+ i ≠j → ({[i := x]} : M A) !! j = None.
Proof. by rewrite lookup_singleton_None. Qed.
-Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]} ≠∅.
+Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]} ≠(∅ : M A).
Proof.
intros Hix. apply (f_equal (!! i)) in Hix.
by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
-Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i := x]} = {[i := y]}.
+Lemma insert_singleton {A} i (x y : A) : <[i:=y]>({[i := x]} : M A) = {[i := y]}.
Proof.
unfold singletonM, map_singleton, insert, map_insert.
by rewrite <-partial_alter_compose.
Qed.
-Lemma alter_singleton {A} (f : A → A) i x : alter f i {[i := x]} = {[i := f x]}.
+Lemma alter_singleton {A} (f : A → A) i x :
+ alter f i ({[i := x]} : M A) = {[i := f x]}.
Proof.
intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
- by rewrite lookup_alter, !lookup_singleton.
- by rewrite lookup_alter_ne, !lookup_singleton_ne.
Qed.
Lemma alter_singleton_ne {A} (f : A → A) i j x :
- i ≠j → alter f i {[j := x]} = {[j := x]}.
+ i ≠j → alter f i ({[j := x]} : M A) = {[j := x]}.
Proof.
intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
Qed.
-Lemma singleton_non_empty {A} i (x : A) : {[i:=x]} ≠∅.
+Lemma singleton_non_empty {A} i (x : A) : {[i:=x]} ≠(∅ : M A).
Proof. apply insert_non_empty. Qed.
-Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = ∅.
+Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = (∅ : M A).
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
Lemma delete_singleton_ne {A} i j (x : A) :
- i ≠j → delete i {[j := x]} = {[j := x]}.
+ i ≠j → delete i ({[j := x]} : M A) = {[j := x]}.
Proof. intro. apply delete_notin. by apply lookup_singleton_ne. Qed.
(** ** Properties of the map operations *)
@@ -621,19 +627,19 @@ Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
Lemma map_fmap_compose {A B C} (f : A → B) (g : B → C) (m : M A) :
g ∘ f <$> m = g <$> f <$> m.
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
-Lemma map_fmap_equiv_ext `{Equiv A, Equiv B} (f1 f2 : A → B) m :
+Lemma map_fmap_equiv_ext `{Equiv A, Equiv B} (f1 f2 : A → B) (m : M A) :
(∀ i x, m !! i = Some x → f1 x ≡ f2 x) → f1 <$> m ≡ f2 <$> m.
Proof.
intros Hi i; rewrite !lookup_fmap.
destruct (m !! i) eqn:?; constructor; eauto.
Qed.
-Lemma map_fmap_ext {A B} (f1 f2 : A → B) m :
+Lemma map_fmap_ext {A B} (f1 f2 : A → B) (m : M A) :
(∀ i x, m !! i = Some x → f1 x = f2 x) → f1 <$> m = f2 <$> m.
Proof.
intros Hi; apply map_eq; intros i; rewrite !lookup_fmap.
by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto.
Qed.
-Lemma omap_ext {A B} (f1 f2 : A → option B) m :
+Lemma omap_ext {A B} (f1 f2 : A → option B) (m : M A) :
(∀ i x, m !! i = Some x → f1 x = f2 x) → omap f1 m = omap f2 m.
Proof.
intros Hi; apply map_eq; intros i; rewrite !lookup_omap.
@@ -670,7 +676,7 @@ Proof. rewrite !elem_of_map_to_list. congruence. Qed.
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
Lemma elem_of_map_of_list_1' {A} (l : list (K * A)) i x :
- (∀ y, (i,y) ∈ l → x = y) → (i,x) ∈ l → map_of_list l !! i = Some x.
+ (∀ y, (i,y) ∈ l → x = y) → (i,x) ∈ l → (map_of_list l : M A) !! i = Some x.
Proof.
induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
setoid_rewrite elem_of_cons.
@@ -680,7 +686,7 @@ Proof.
- rewrite lookup_insert_ne by done; eauto.
Qed.
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
- NoDup (l.*1) → (i,x) ∈ l → map_of_list l !! i = Some x.
+ NoDup (l.*1) → (i,x) ∈ l → (map_of_list l : M A) !! i = Some x.
Proof.
intros ? Hx; apply elem_of_map_of_list_1'; eauto using NoDup_fmap_fst.
intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
@@ -688,7 +694,7 @@ Proof.
by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
- map_of_list l !! i = Some x → (i,x) ∈ l.
+ (map_of_list l : M A) !! i = Some x → (i,x) ∈ l.
Proof.
induction l as [|[j y] l IH]; simpl; [by rewrite lookup_empty|].
rewrite elem_of_cons. destruct (decide (i = j)) as [->|];
@@ -696,20 +702,20 @@ Proof.
Qed.
Lemma elem_of_map_of_list' {A} (l : list (K * A)) i x :
(∀ x', (i,x) ∈ l → (i,x') ∈ l → x = x') →
- (i,x) ∈ l ↔ map_of_list l !! i = Some x.
+ (i,x) ∈ l ↔ (map_of_list l : M A) !! i = Some x.
Proof. split; auto using elem_of_map_of_list_1', elem_of_map_of_list_2. Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
- NoDup (l.*1) → (i,x) ∈ l ↔ map_of_list l !! i = Some x.
+ NoDup (l.*1) → (i,x) ∈ l ↔ (map_of_list l : M A) !! i = Some x.
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
- i ∉ l.*1 → map_of_list l !! i = None.
+ i ∉ l.*1 → (map_of_list l : M A) !! i = None.
Proof.
rewrite elem_of_list_fmap, eq_None_not_Some. intros Hi [x ?]; destruct Hi.
exists (i,x); simpl; auto using elem_of_map_of_list_2.
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
- map_of_list l !! i = None → i ∉ l.*1.
+ (map_of_list l : M A) !! i = None → i ∉ l.*1.
Proof.
induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
rewrite elem_of_cons. destruct (decide (i = j)); simplify_eq.
@@ -717,16 +723,17 @@ Proof.
- by rewrite lookup_insert_ne; intuition.
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
- i ∉ l.*1 ↔ map_of_list l !! i = None.
+ i ∉ l.*1 ↔ (map_of_list l : M A) !! i = None.
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
- NoDup (l1.*1) → l1 ≡ₚ l2 → map_of_list l1 = map_of_list l2.
+ NoDup (l1.*1) → l1 ≡ₚ l2 → (map_of_list l1 : M A) = map_of_list l2.
Proof.
intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
Qed.
Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
- NoDup (l1.*1) → NoDup (l2.*1) → map_of_list l1 = map_of_list l2 → l1 ≡ₚ l2.
+ NoDup (l1.*1) → NoDup (l2.*1) →
+ (map_of_list l1 : M A) = map_of_list l2 → l1 ≡ₚ l2.
Proof.
intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
@@ -753,13 +760,13 @@ Proof.
auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
-Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = ∅.
+Lemma map_of_list_nil {A} : map_of_list [] = (∅ : M A).
Proof. done. Qed.
Lemma map_of_list_cons {A} (l : list (K * A)) i x :
- map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l).
+ map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l : M A).
Proof. done. Qed.
Lemma map_of_list_fmap {A B} (f : A → B) l :
- map_of_list (prod_map id f <$> l) = f <$> map_of_list l.
+ map_of_list (prod_map id f <$> l) = f <$> (map_of_list l : M A).
Proof.
induction l as [|[i x] l IH]; csimpl; rewrite ?fmap_empty; auto.
rewrite <-map_of_list_cons; simpl. by rewrite IH, <-fmap_insert.
@@ -780,7 +787,8 @@ Proof.
rewrite elem_of_map_to_list in Hlookup. congruence.
- by rewrite !map_of_to_list.
Qed.
-Lemma map_to_list_singleton {A} i (x : A) : map_to_list {[i:=x]} = [(i,x)].
+Lemma map_to_list_singleton {A} i (x : A) :
+ map_to_list ({[i:=x]} : M A) = [(i,x)].
Proof.
apply Permutation_singleton. unfold singletonM, map_singleton.
by rewrite map_to_list_insert, map_to_list_empty by auto using lookup_empty.
@@ -792,7 +800,7 @@ Proof.
intros; apply NoDup_submseteq; auto using NoDup_map_to_list.
intros [i x]. rewrite !elem_of_map_to_list; eauto using lookup_weaken.
Qed.
-Lemma map_to_list_fmap {A B} (f : A → B) m :
+Lemma map_to_list_fmap {A B} (f : A → B) (m : M A) :
map_to_list (f <$> m) ≡ₚ prod_map id f <$> map_to_list m.
Proof.
assert (NoDup ((prod_map id f <$> map_to_list m).*1)).
@@ -836,7 +844,7 @@ Proof.
Defined.
(** Properties of the imap function *)
-Lemma lookup_imap {A B} (f : K → A → option B) m i :
+Lemma lookup_imap {A B} (f : K → A → option B) (m : M A) i :
map_imap f m !! i = m !! i ≫= f i.
Proof.
unfold map_imap; destruct (m !! i ≫= f i) as [y|] eqn:Hi; simpl.
@@ -856,9 +864,9 @@ Qed.
Section map_of_to_collection.
Context {A : Type} `{FinCollection B C}.
- Lemma lookup_map_of_collection (f : B → K * A) Y i x :
+ Lemma lookup_map_of_collection (f : B → K * A) (Y : C) i x :
(∀ y y', y ∈ Y → y' ∈ Y → (f y).1 = (f y').1 → y = y') →
- map_of_collection f Y !! i = Some x ↔ ∃ y, y ∈ Y ∧ f y = (i,x).
+ (map_of_collection f Y : M A) !! i = Some x ↔ ∃ y, y ∈ Y ∧ f y = (i,x).
Proof.
intros Hinj. assert (∀ x',
(i, x) ∈ f <$> elements Y → (i, x') ∈ f <$> elements Y → x = x').
@@ -870,14 +878,15 @@ Section map_of_to_collection.
Qed.
Lemma elem_of_map_to_collection (f : K → A → B) (m : M A) (y : B) :
- y ∈ map_to_collection f m ↔ ∃ i x, m !! i = Some x ∧ f i x = y.
+ y ∈ map_to_collection (C:=C) f m ↔ ∃ i x, m !! i = Some x ∧ f i x = y.
Proof.
unfold map_to_collection; simpl.
rewrite elem_of_of_list, elem_of_list_fmap. split.
- intros ([i x] & ? & ?%elem_of_map_to_list); eauto.
- intros (i&x&?&?). exists (i,x). by rewrite elem_of_map_to_list.
Qed.
- Lemma map_to_collection_empty (f : K → A → B) : map_to_collection f ∅ = ∅.
+ Lemma map_to_collection_empty (f : K → A → B) :
+ map_to_collection f (∅ : M A) = (∅ : C).
Proof. unfold map_to_collection; simpl. by rewrite map_to_list_empty. Qed.
Lemma map_to_collection_insert (f : K → A → B)(m : M A) i x :
m !! i = None →
@@ -885,7 +894,7 @@ Section map_of_to_collection.
Proof.
intros. unfold map_to_collection; simpl. by rewrite map_to_list_insert.
Qed.
- Lemma map_to_collection_insert_L `{!LeibnizEquiv C} (f : K → A → B) m i x :
+ Lemma map_to_collection_insert_L `{!LeibnizEquiv C} (f : K → A → B) (m : M A) i x :
m !! i = None →
map_to_collection (C:=C) f (<[i:=x]>m) = {[f i x]} ∪ map_to_collection f m.
Proof. unfold_leibniz. apply map_to_collection_insert. Qed.
@@ -893,7 +902,7 @@ End map_of_to_collection.
Lemma lookup_map_of_collection_id `{FinCollection (K * A) C} (X : C) i x :
(∀ i y y', (i,y) ∈ X → (i,y') ∈ X → y = y') →
- map_of_collection id X !! i = Some x ↔ (i,x) ∈ X.
+ (map_of_collection id X : M A) !! i = Some x ↔ (i,x) ∈ X.
Proof.
intros. etrans; [apply lookup_map_of_collection|naive_solver].
intros [] [] ???; simplify_eq/=; eauto with f_equal.
@@ -996,6 +1005,7 @@ Qed.
(** ** Properties of the [map_Forall] predicate *)
Section map_Forall.
Context {A} (P : K → A → Prop).
+Implicit Types m : M A.
Lemma map_Forall_to_list m : map_Forall P m ↔ Forall (curry P) (map_to_list m).
Proof.
@@ -1003,7 +1013,7 @@ Proof.
- intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x).
- intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)).
Qed.
-Lemma map_Forall_empty : map_Forall P ∅.
+Lemma map_Forall_empty : map_Forall P (∅ : M A).
Proof. intros i x. by rewrite lookup_empty. Qed.
Lemma map_Forall_impl (Q : K → A → Prop) m :
map_Forall P m → (∀ i x, P i x → Q i x) → map_Forall Q m.
@@ -1052,12 +1062,14 @@ End map_Forall.
(** ** Properties of the [merge] operation *)
Section merge.
Context {A} (f : option A → option A → option A) `{!DiagNone f}.
-Global Instance: LeftId (=) None f → LeftId (=) ∅ (merge f).
+Implicit Types m : M A.
+
+Global Instance: LeftId (=) None f → LeftId (=) (∅ : M A) (merge f).
Proof.
intros ??. apply map_eq. intros.
by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f).
Qed.
-Global Instance: RightId (=) None f → RightId (=) ∅ (merge f).
+Global Instance: RightId (=) None f → RightId (=) (∅ : M A) (merge f).
Proof.
intros ??. apply map_eq. intros.
by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
@@ -1066,34 +1078,34 @@ Lemma merge_comm m1 m2 :
(∀ i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) →
merge f m1 m2 = merge f m2 m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
-Global Instance merge_comm' : Comm (=) f → Comm (=) (merge f).
+Global Instance merge_comm' : Comm (=) f → Comm (=) (merge (M:=M) f).
Proof. intros ???. apply merge_comm. intros. by apply (comm f). Qed.
Lemma merge_assoc m1 m2 m3 :
(∀ i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) =
f (f (m1 !! i) (m2 !! i)) (m3 !! i)) →
merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
-Global Instance merge_assoc' : Assoc (=) f → Assoc (=) (merge f).
+Global Instance merge_assoc' : Assoc (=) f → Assoc (=) (merge (M:=M) f).
Proof. intros ????. apply merge_assoc. intros. by apply (assoc_L f). Qed.
Lemma merge_idemp m1 :
(∀ i, f (m1 !! i) (m1 !! i) = m1 !! i) → merge f m1 m1 = m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
-Global Instance merge_idemp' : IdemP (=) f → IdemP (=) (merge f).
+Global Instance merge_idemp' : IdemP (=) f → IdemP (=) (merge (M:=M) f).
Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed.
End merge.
Section more_merge.
Context {A B C} (f : option A → option B → option C) `{!DiagNone f}.
-Lemma merge_Some m1 m2 m :
+Lemma merge_Some (m1 : M A) (m2 : M B) (m : M C) :
(∀ i, m !! i = f (m1 !! i) (m2 !! i)) ↔ merge f m1 m2 = m.
Proof.
split; [|intros <-; apply (lookup_merge _) ].
intros Hlookup. apply map_eq; intros. rewrite Hlookup. apply (lookup_merge _).
Qed.
-Lemma merge_empty : merge f ∅ ∅ = ∅.
+Lemma merge_empty : merge f (∅ : M A) (∅ : M B) = ∅.
Proof. apply map_eq. intros. by rewrite !(lookup_merge f), !lookup_empty. Qed.
-Lemma partial_alter_merge g g1 g2 m1 m2 i :
+Lemma partial_alter_merge g g1 g2 (m1 : M A) (m2 : M B) i :
g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (g2 (m2 !! i)) →
partial_alter g i (merge f m1 m2) =
merge f (partial_alter g1 i m1) (partial_alter g2 i m2).
@@ -1102,7 +1114,7 @@ Proof.
- by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
-Lemma partial_alter_merge_l g g1 m1 m2 i :
+Lemma partial_alter_merge_l g g1 (m1 : M A) (m2 : M B) i :
g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (m2 !! i) →
partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) m2.
Proof.
@@ -1110,7 +1122,7 @@ Proof.
- by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
-Lemma partial_alter_merge_r g g2 m1 m2 i :
+Lemma partial_alter_merge_r g g2 (m1 : M A) (m2 : M B) i :
g (f (m1 !! i) (m2 !! i)) = f (m1 !! i) (g2 (m2 !! i)) →
partial_alter g i (merge f m1 m2) = merge f m1 (partial_alter g2 i m2).
Proof.
@@ -1118,37 +1130,39 @@ Proof.
- by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
-Lemma insert_merge m1 m2 i x y z :
+Lemma insert_merge (m1 : M A) (m2 : M B) i x y z :
f (Some y) (Some z) = Some x →
<[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) (<[i:=z]>m2).
Proof. by intros; apply partial_alter_merge. Qed.
Lemma merge_singleton i x y z :
- f (Some y) (Some z) = Some x → merge f {[i := y]} {[i := z]} = {[i := x]}.
+ f (Some y) (Some z) = Some x →
+ merge f ({[i := y]} : M A) ({[i := z]} : M B) = {[i := x]}.
Proof.
intros. by erewrite <-!insert_empty, <-insert_merge, merge_empty by eauto.
Qed.
-Lemma insert_merge_l m1 m2 i x y :
+Lemma insert_merge_l (m1 : M A) (m2 : M B) i x y :
f (Some y) (m2 !! i) = Some x →
<[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) m2.
Proof. by intros; apply partial_alter_merge_l. Qed.
-Lemma insert_merge_r m1 m2 i x z :
+Lemma insert_merge_r (m1 : M A) (m2 : M B) i x z :
f (m1 !! i) (Some z) = Some x →
<[i:=x]>(merge f m1 m2) = merge f m1 (<[i:=z]>m2).
Proof. by intros; apply partial_alter_merge_r. Qed.
End more_merge.
(** Properties of the zip_with function *)
-Lemma map_lookup_zip_with {A B C} (f : A → B → C) m1 m2 i :
+Lemma map_lookup_zip_with {A B C} (f : A → B → C) (m1 : M A) (m2 : M B) i :
map_zip_with f m1 m2 !! i = x ↠m1 !! i; y ↠m2 !! i; Some (f x y).
Proof.
unfold map_zip_with. rewrite lookup_merge by done.
by destruct (m1 !! i), (m2 !! i).
Qed.
-Lemma map_zip_with_empty {A B C} (f : A → B → C) : map_zip_with f ∅ ∅ = ∅.
+Lemma map_zip_with_empty {A B C} (f : A → B → C) :
+ map_zip_with f (∅ : M A) (∅ : M B) = ∅.
Proof. unfold map_zip_with. by rewrite merge_empty by done. Qed.
-Lemma map_insert_zip_with {A B C} (f : A → B → C) m1 m2 i x y z :
+Lemma map_insert_zip_with {A B C} (f : A → B → C) (m1 : M A) (m2 : M B) i x y z :
f y z = x →
<[i:=x]>(map_zip_with f m1 m2) = map_zip_with f (<[i:=y]>m1) (<[i:=z]>m2).
Proof.
@@ -1157,7 +1171,7 @@ Proof.
Qed.
Lemma map_zip_with_fmap {A' A B' B C} (f : A → B → C)
- (g1 : A' → A) (g2 : B' → B) m1 m2 :
+ (g1 : A' → A) (g2 : B' → B) (m1 : M A') (m2 : M B') :
map_zip_with f (g1 <$> m1) (g2 <$> m2) = map_zip_with (λ x y, f (g1 x) (g2 y)) m1 m2.
Proof.
apply map_eq; intro i. rewrite !map_lookup_zip_with, !lookup_fmap.
@@ -1165,13 +1179,14 @@ Proof.
Qed.
Lemma map_zip_with_fmap_1 {A' A B C} (f : A → B → C)
- (g : A' → A) m1 m2 :
+ (g : A' → A) (m1 : M A') (m2 : M B) :
map_zip_with f (g <$> m1) m2 = map_zip_with (λ x y, f (g x) y) m1 m2.
Proof.
rewrite <- (map_fmap_id m2) at 1. by rewrite map_zip_with_fmap.
Qed.
-Lemma map_zip_with_fmap_2 {A B' B C} (f : A → B → C) (g : B' → B) m1 m2 :
+Lemma map_zip_with_fmap_2 {A B' B C} (f : A → B → C)
+ (g : B' → B) (m1 : M A) (m2 : M B') :
map_zip_with f m1 (g <$> m2) = map_zip_with (λ x y, f x (g y)) m1 m2.
Proof.
rewrite <- (map_fmap_id m1) at 1. by rewrite map_zip_with_fmap.
@@ -1184,14 +1199,15 @@ Proof.
by destruct (m1 !! i), (m2 !! i).
Qed.
-Lemma map_zip_with_map_zip {A B C} (f : A → B → C) m1 m2 :
+Lemma map_zip_with_map_zip {A B C} (f : A → B → C) (m1 : M A) (m2 : M B) :
map_zip_with f m1 m2 = curry f <$> map_zip m1 m2.
Proof.
apply map_eq; intro i. rewrite lookup_fmap, !map_lookup_zip_with.
by destruct (m1 !! i), (m2 !! i).
Qed.
-Lemma map_fmap_zip {A' A B' B} (g1 : A' → A) (g2 : B' → B) m1 m2 :
+Lemma map_fmap_zip {A' A B' B} (g1 : A' → A)
+ (g2 : B' → B) (m1 : M A') (m2 : M B') :
map_zip (fmap g1 m1) (fmap g2 m2) = prod_map g1 g2 <$> map_zip m1 m2.
Proof.
rewrite map_zip_with_fmap, map_zip_with_map_zip.
@@ -1223,7 +1239,7 @@ Proof.
by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_relation_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x),
- ∀ y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2).
+ ∀ y, Decision (Q y)} (m1 : M A) (m2 : M B) : Decision (map_relation R P Q m1 m2).
Proof.
refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
abstract by rewrite map_relation_alt.
@@ -1316,6 +1332,7 @@ Proof. symmetry. by apply map_disjoint_delete_l. Qed.
(** ** Properties of the [union_with] operation *)
Section union_with.
Context {A} (f : A → A → option A).
+Implicit Types m : M A.
Lemma lookup_union_with m1 m2 i :
union_with f m1 m2 !! i = union_with f (m1 !! i) (m2 !! i).
@@ -1659,10 +1676,10 @@ Lemma map_disjoint_of_list_zip_r_2 {A} (m : M A) is xs :
Proof. intro. by rewrite map_disjoint_of_list_zip_r. Qed.
(** ** Properties of the [intersection_with] operation *)
-Lemma lookup_intersection_with {A} (f : A → A → option A) m1 m2 i :
+Lemma lookup_intersection_with {A} (f : A → A → option A) (m1 m2 : M A) i :
intersection_with f m1 m2 !! i = intersection_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-(lookup_merge _). Qed.
-Lemma lookup_intersection_with_Some {A} (f : A → A → option A) m1 m2 i z :
+Lemma lookup_intersection_with_Some {A} (f : A → A → option A) (m1 m2 : M A) i z :
intersection_with f m1 m2 !! i = Some z ↔
(∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ f x y = Some z).
Proof.
@@ -1685,10 +1702,10 @@ Proof.
Qed.
(** ** Properties of the [difference_with] operation *)
-Lemma lookup_difference_with {A} (f : A → A → option A) m1 m2 i :
+Lemma lookup_difference_with {A} (f : A → A → option A) (m1 m2 : M A) i :
difference_with f m1 m2 !! i = difference_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-lookup_merge by done. Qed.
-Lemma lookup_difference_with_Some {A} (f : A → A → option A) m1 m2 i z :
+Lemma lookup_difference_with_Some {A} (f : A → A → option A) (m1 m2 : M A) i z :
difference_with f m1 m2 !! i = Some z ↔
(m1 !! i = Some z ∧ m2 !! i = None) ∨
(∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ f x y = Some z).
diff --git a/theories/gmap.v b/theories/gmap.v
index fe7ddf772f5a9a8590c9210a543d6064d110f154..874fff964215e31b974a8444de62534ff9c12d3b 100644
--- a/theories/gmap.v
+++ b/theories/gmap.v
@@ -136,8 +136,10 @@ Definition gmap_uncurry `{Countable K1, Countable K2} {A} :
Section curry_uncurry.
Context `{Countable K1, Countable K2} {A : Type}.
+ (* FIXME: the type annotations `option (gmap K2 A)` are silly. Maybe these are
+ a consequence of Coq bug #5735 *)
Lemma lookup_gmap_curry (m : gmap K1 (gmap K2 A)) i j :
- gmap_curry m !! (i,j) = m !! i ≫= (!! j).
+ gmap_curry m !! (i,j) = (m !! i : option (gmap K2 A)) ≫= (!! j).
Proof.
apply (map_fold_ind (λ mr m, mr !! (i,j) = m !! i ≫= (!! j))).
{ by rewrite !lookup_empty. }
@@ -154,7 +156,7 @@ Section curry_uncurry.
Qed.
Lemma lookup_gmap_uncurry (m : gmap (K1 * K2) A) i j :
- gmap_uncurry m !! i ≫= (!! j) = m !! (i, j).
+ (gmap_uncurry m !! i : option (gmap K2 A)) ≫= (!! j) = m !! (i, j).
Proof.
apply (map_fold_ind (λ mr m, mr !! i ≫= (!! j) = m !! (i, j))).
{ by rewrite !lookup_empty. }
@@ -229,7 +231,7 @@ Qed.
(* This is pretty ad-hoc and just for the case of [gset positive]. We need a
notion of countable non-finite types to generalize this. *)
Instance gset_positive_fresh : Fresh positive (gset positive) := λ X,
- let 'Mapset (GMap m _) := X in fresh (dom _ m).
+ let 'Mapset (GMap m _) := X in fresh (dom Pset m).
Instance gset_positive_fresh_spec : FreshSpec positive (gset positive).
Proof.
split.
diff --git a/theories/gmultiset.v b/theories/gmultiset.v
index 084dfdca5b609518591529f7fa862744c284cdec..f61416e9627f0953bcdbdb3f6b78942a2991b04b 100644
--- a/theories/gmultiset.v
+++ b/theories/gmultiset.v
@@ -25,27 +25,27 @@ Section definitions.
Definition multiplicity (x : A) (X : gmultiset A) : nat :=
match gmultiset_car X !! x with Some n => S n | None => 0 end.
- Instance gmultiset_elem_of : ElemOf A (gmultiset A) := λ x X,
+ Global Instance gmultiset_elem_of : ElemOf A (gmultiset A) := λ x X,
0 < multiplicity x X.
- Instance gmultiset_subseteq : SubsetEq (gmultiset A) := λ X Y, ∀ x,
+ Global Instance gmultiset_subseteq : SubsetEq (gmultiset A) := λ X Y, ∀ x,
multiplicity x X ≤ multiplicity x Y.
- Instance gmultiset_elements : Elements A (gmultiset A) := λ X,
+ Global Instance gmultiset_elements : Elements A (gmultiset A) := λ X,
let (X) := X in '(x,n) ↠map_to_list X; replicate (S n) x.
- Instance gmultiset_size : Size (gmultiset A) := length ∘ elements.
+ Global Instance gmultiset_size : Size (gmultiset A) := length ∘ elements.
- Instance gmultiset_empty : Empty (gmultiset A) := GMultiSet ∅.
- Instance gmultiset_singleton : Singleton A (gmultiset A) := λ x,
+ Global Instance gmultiset_empty : Empty (gmultiset A) := GMultiSet ∅.
+ Global Instance gmultiset_singleton : Singleton A (gmultiset A) := λ x,
GMultiSet {[ x := 0 ]}.
- Instance gmultiset_union : Union (gmultiset A) := λ X Y,
+ Global Instance gmultiset_union : Union (gmultiset A) := λ X Y,
let (X) := X in let (Y) := Y in
GMultiSet $ union_with (λ x y, Some (S (x + y))) X Y.
- Instance gmultiset_difference : Difference (gmultiset A) := λ X Y,
+ Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y,
let (X) := X in let (Y) := Y in
GMultiSet $ difference_with (λ x y,
let z := x - y in guard (0 < z); Some (pred z)) X Y.
- Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X,
+ Global Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X,
let (X) := X in dom _ X.
End definitions.
@@ -54,27 +54,6 @@ Typeclasses Opaque gmultiset_elements gmultiset_size gmultiset_empty.
Typeclasses Opaque gmultiset_singleton gmultiset_union gmultiset_difference.
Typeclasses Opaque gmultiset_dom.
-(** These instances are declared using [Hint Extern] to avoid too
-eager type class search. *)
-Hint Extern 1 (ElemOf _ (gmultiset _)) =>
- eapply @gmultiset_elem_of : typeclass_instances.
-Hint Extern 1 (SubsetEq (gmultiset _)) =>
- eapply @gmultiset_subseteq : typeclass_instances.
-Hint Extern 1 (Empty (gmultiset _)) =>
- eapply @gmultiset_empty : typeclass_instances.
-Hint Extern 1 (Singleton _ (gmultiset _)) =>
- eapply @gmultiset_singleton : typeclass_instances.
-Hint Extern 1 (Union (gmultiset _)) =>
- eapply @gmultiset_union : typeclass_instances.
-Hint Extern 1 (Difference (gmultiset _)) =>
- eapply @gmultiset_difference : typeclass_instances.
-Hint Extern 1 (Elements _ (gmultiset _)) =>
- eapply @gmultiset_elements : typeclass_instances.
-Hint Extern 1 (Size (gmultiset _)) =>
- eapply @gmultiset_size : typeclass_instances.
-Hint Extern 1 (Dom (gmultiset _) _) =>
- eapply @gmultiset_dom : typeclass_instances.
-
Section lemmas.
Context `{Countable A}.
Implicit Types x y : A.
diff --git a/theories/hashset.v b/theories/hashset.v
index 59c5506ba5c5527585b551223a16c46cb5f8029c..20a7d4ae07c1507310e459cdfa5cd8c0ebc53d52 100644
--- a/theories/hashset.v
+++ b/theories/hashset.v
@@ -18,18 +18,18 @@ Arguments hashset_car {_ _} _ : assert.
Section hashset.
Context `{EqDecision A} (hash : A → Z).
-Instance hashset_elem_of: ElemOf A (hashset hash) := λ x m, ∃ l,
+Global Instance hashset_elem_of: ElemOf A (hashset hash) := λ x m, ∃ l,
hashset_car m !! hash x = Some l ∧ x ∈ l.
-Program Instance hashset_empty: Empty (hashset hash) := Hashset ∅ _.
+Global Program Instance hashset_empty: Empty (hashset hash) := Hashset ∅ _.
Next Obligation. by intros n X; simpl_map. Qed.
-Program Instance hashset_singleton: Singleton A (hashset hash) := λ x,
+Global Program Instance hashset_singleton: Singleton A (hashset hash) := λ x,
Hashset {[ hash x := [x] ]} _.
Next Obligation.
intros x n l [<- <-]%lookup_singleton_Some.
rewrite Forall_singleton; auto using NoDup_singleton.
Qed.
-Program Instance hashset_union: Union (hashset hash) := λ m1 m2,
+Global Program Instance hashset_union: Union (hashset hash) := λ m1 m2,
let (m1,Hm1) := m1 in let (m2,Hm2) := m2 in
Hashset (union_with (λ l k, Some (list_union l k)) m1 m2) _.
Next Obligation.
@@ -38,7 +38,7 @@ Next Obligation.
split; [apply Forall_list_union|apply NoDup_list_union];
first [by eapply Hm1; eauto | by eapply Hm2; eauto].
Qed.
-Program Instance hashset_intersection: Intersection (hashset hash) := λ m1 m2,
+Global Program Instance hashset_intersection: Intersection (hashset hash) := λ m1 m2,
let (m1,Hm1) := m1 in let (m2,Hm2) := m2 in
Hashset (intersection_with (λ l k,
let l' := list_intersection l k in guard (l' ≠[]); Some l') m1 m2) _.
@@ -48,7 +48,7 @@ Next Obligation.
split; [apply Forall_list_intersection|apply NoDup_list_intersection];
first [by eapply Hm1; eauto | by eapply Hm2; eauto].
Qed.
-Program Instance hashset_difference: Difference (hashset hash) := λ m1 m2,
+Global Program Instance hashset_difference: Difference (hashset hash) := λ m1 m2,
let (m1,Hm1) := m1 in let (m2,Hm2) := m2 in
Hashset (difference_with (λ l k,
let l' := list_difference l k in guard (l' ≠[]); Some l') m1 m2) _.
@@ -58,10 +58,10 @@ Next Obligation.
split; [apply Forall_list_difference|apply NoDup_list_difference];
first [by eapply Hm1; eauto | by eapply Hm2; eauto].
Qed.
-Instance hashset_elems: Elements A (hashset hash) := λ m,
+Global Instance hashset_elements: Elements A (hashset hash) := λ m,
map_to_list (hashset_car m) ≫= snd.
-Global Instance: FinCollection A (hashset hash).
+Global Instance hashset_fin_collection : FinCollection A (hashset hash).
Proof.
split; [split; [split| |]| |].
- intros ? (?&?&?); simplify_map_eq/=.
@@ -98,13 +98,13 @@ Proof.
assert (x ∈ list_difference l k) by (by rewrite elem_of_list_difference).
exists (list_difference l k); split; [right; exists l,k|]; split_and?; auto.
by rewrite option_guard_True by eauto using elem_of_not_nil.
- - unfold elem_of at 2, hashset_elem_of, elements, hashset_elems.
+ - unfold elem_of at 2, hashset_elem_of, elements, hashset_elements.
intros [m Hm] x; simpl. setoid_rewrite elem_of_list_bind. split.
{ intros ([n l]&Hx&Hn); simpl in *; rewrite elem_of_map_to_list in Hn.
cut (hash x = n); [intros <-; eauto|].
eapply (Forall_forall (λ x, hash x = n) l); eauto. eapply Hm; eauto. }
intros (l&?&?). exists (hash x, l); simpl. by rewrite elem_of_map_to_list.
- - unfold elements, hashset_elems. intros [m Hm]; simpl.
+ - unfold elements, hashset_elements. intros [m Hm]; simpl.
rewrite map_Forall_to_list in Hm. generalize (NoDup_fst_map_to_list m).
induction Hm as [|[n l] m' [??]];
csimpl; inversion_clear 1 as [|?? Hn]; [constructor|].
@@ -120,23 +120,6 @@ End hashset.
Typeclasses Opaque hashset_elem_of.
-(** These instances are declared using [Hint Extern] to avoid too
-eager type class search. *)
-Hint Extern 1 (ElemOf _ (hashset _)) =>
- eapply @hashset_elem_of : typeclass_instances.
-Hint Extern 1 (Empty (hashset _)) =>
- eapply @hashset_empty : typeclass_instances.
-Hint Extern 1 (Singleton _ (hashset _)) =>
- eapply @hashset_singleton : typeclass_instances.
-Hint Extern 1 (Union (hashset _)) =>
- eapply @hashset_union : typeclass_instances.
-Hint Extern 1 (Intersection (hashset _)) =>
- eapply @hashset_intersection : typeclass_instances.
-Hint Extern 1 (Difference (hashset _)) =>
- eapply @hashset_difference : typeclass_instances.
-Hint Extern 1 (Elements _ (hashset _)) =>
- eapply @hashset_elems : typeclass_instances.
-
Section remove_duplicates.
Context `{EqDecision A} (hash : A → Z).
diff --git a/theories/list.v b/theories/list.v
index 5f3bf7840e3ac26ab2cba52be7be27b8e0a30fc1..a258bea8e2959d1c4d7010aeb01aa51a708dcef4 100644
--- a/theories/list.v
+++ b/theories/list.v
@@ -311,7 +311,7 @@ Instance list_subseteq {A} : SubsetEq (list A) := λ l1 l2, ∀ x, x ∈ l1 →
Section list_set.
Context `{dec : EqDecision A}.
- Global Instance elem_of_list_dec (x : A) : ∀ l, Decision (x ∈ l).
+ Global Instance elem_of_list_dec (x : A) : ∀ l : list A, Decision (x ∈ l).
Proof.
refine (
fix go l :=
@@ -2857,6 +2857,7 @@ Proof. induction l; f_equal/=; auto. Qed.
Section fmap.
Context {A B : Type} (f : A → B).
+ Implicit Types l : list A.
Lemma list_fmap_compose {C} (g : B → C) l : g ∘ f <$> l = g <$> f <$> l.
Proof. induction l; f_equal/=; auto. Qed.
@@ -2975,24 +2976,24 @@ Section fmap.
Lemma Exists_fmap (P : B → Prop) l : Exists P (f <$> l) ↔ Exists (P ∘ f) l.
Proof. split; induction l; inversion 1; constructor; by auto. Qed.
- Lemma Forall2_fmap_l {C} (P : B → C → Prop) l1 l2 :
- Forall2 P (f <$> l1) l2 ↔ Forall2 (P ∘ f) l1 l2.
+ Lemma Forall2_fmap_l {C} (P : B → C → Prop) l k :
+ Forall2 P (f <$> l) k ↔ Forall2 (P ∘ f) l k.
Proof.
- split; revert l2; induction l1; inversion_clear 1; constructor; auto.
+ split; revert k; induction l; inversion_clear 1; constructor; auto.
Qed.
- Lemma Forall2_fmap_r {C} (P : C → B → Prop) l1 l2 :
- Forall2 P l1 (f <$> l2) ↔ Forall2 (λ x, P x ∘ f) l1 l2.
+ Lemma Forall2_fmap_r {C} (P : C → B → Prop) k l :
+ Forall2 P k (f <$> l) ↔ Forall2 (λ x, P x ∘ f) k l.
Proof.
- split; revert l1; induction l2; inversion_clear 1; constructor; auto.
+ split; revert k; induction l; inversion_clear 1; constructor; auto.
Qed.
- Lemma Forall2_fmap_1 {C D} (g : C → D) (P : B → D → Prop) l1 l2 :
- Forall2 P (f <$> l1) (g <$> l2) → Forall2 (λ x1 x2, P (f x1) (g x2)) l1 l2.
- Proof. revert l2; induction l1; intros [|??]; inversion_clear 1; auto. Qed.
- Lemma Forall2_fmap_2 {C D} (g : C → D) (P : B → D → Prop) l1 l2 :
- Forall2 (λ x1 x2, P (f x1) (g x2)) l1 l2 → Forall2 P (f <$> l1) (g <$> l2).
+ Lemma Forall2_fmap_1 {C D} (g : C → D) (P : B → D → Prop) l k :
+ Forall2 P (f <$> l) (g <$> k) → Forall2 (λ x1 x2, P (f x1) (g x2)) l k.
+ Proof. revert k; induction l; intros [|??]; inversion_clear 1; auto. Qed.
+ Lemma Forall2_fmap_2 {C D} (g : C → D) (P : B → D → Prop) l k :
+ Forall2 (λ x1 x2, P (f x1) (g x2)) l k → Forall2 P (f <$> l) (g <$> k).
Proof. induction 1; csimpl; auto. Qed.
- Lemma Forall2_fmap {C D} (g : C → D) (P : B → D → Prop) l1 l2 :
- Forall2 P (f <$> l1) (g <$> l2) ↔ Forall2 (λ x1 x2, P (f x1) (g x2)) l1 l2.
+ Lemma Forall2_fmap {C D} (g : C → D) (P : B → D → Prop) l k :
+ Forall2 P (f <$> l) (g <$> k) ↔ Forall2 (λ x1 x2, P (f x1) (g x2)) l k.
Proof. split; auto using Forall2_fmap_1, Forall2_fmap_2. Qed.
Lemma list_fmap_bind {C} (g : B → list C) l : (f <$> l) ≫= g = l ≫= g ∘ f.
@@ -3081,7 +3082,7 @@ Section ret_join.
Lemma elem_of_list_ret (x y : A) : x ∈ @mret list _ A y ↔ x = y.
Proof. apply elem_of_list_singleton. Qed.
Lemma elem_of_list_join (x : A) (ls : list (list A)) :
- x ∈ mjoin ls ↔ ∃ l, x ∈ l ∧ l ∈ ls.
+ x ∈ mjoin ls ↔ ∃ l : list A, x ∈ l ∧ l ∈ ls.
Proof. by rewrite list_join_bind, elem_of_list_bind. Qed.
Lemma join_nil (ls : list (list A)) : mjoin ls = [] ↔ Forall (= []) ls.
Proof.
diff --git a/theories/listset.v b/theories/listset.v
index 4649ede5db3ee1661003324e158d57c2c02ec115..824cbc4513a873ad720018ca11c29baf92cc3f12 100644
--- a/theories/listset.v
+++ b/theories/listset.v
@@ -12,14 +12,14 @@ Arguments Listset {_} _ : assert.
Section listset.
Context {A : Type}.
-Instance listset_elem_of: ElemOf A (listset A) := λ x l, x ∈ listset_car l.
-Instance listset_empty: Empty (listset A) := Listset [].
-Instance listset_singleton: Singleton A (listset A) := λ x, Listset [x].
-Instance listset_union: Union (listset A) := λ l k,
+Global Instance listset_elem_of: ElemOf A (listset A) := λ x l, x ∈ listset_car l.
+Global Instance listset_empty: Empty (listset A) := Listset [].
+Global Instance listset_singleton: Singleton A (listset A) := λ x, Listset [x].
+Global Instance listset_union: Union (listset A) := λ l k,
let (l') := l in let (k') := k in Listset (l' ++ k').
Global Opaque listset_singleton listset_empty.
-Global Instance: SimpleCollection A (listset A).
+Global Instance listset_simple_collection : SimpleCollection A (listset A).
Proof.
split.
- by apply not_elem_of_nil.
@@ -40,20 +40,21 @@ Defined.
Context `{!EqDecision A}.
-Instance listset_intersection: Intersection (listset A) := λ l k,
+Global Instance listset_intersection: Intersection (listset A) := λ l k,
let (l') := l in let (k') := k in Listset (list_intersection l' k').
-Instance listset_difference: Difference (listset A) := λ l k,
+Global Instance listset_difference: Difference (listset A) := λ l k,
let (l') := l in let (k') := k in Listset (list_difference l' k').
-Instance: Collection A (listset A).
+Instance listset_collection: Collection A (listset A).
Proof.
split.
- apply _.
- intros [?] [?]. apply elem_of_list_intersection.
- intros [?] [?]. apply elem_of_list_difference.
Qed.
-Instance listset_elems: Elements A (listset A) := remove_dups ∘ listset_car.
-Global Instance: FinCollection A (listset A).
+Global Instance listset_elements: Elements A (listset A) :=
+ remove_dups ∘ listset_car.
+Global Instance listset_fin_collection : FinCollection A (listset A).
Proof.
split.
- apply _.
@@ -62,23 +63,6 @@ Proof.
Qed.
End listset.
-(** These instances are declared using [Hint Extern] to avoid too
-eager type class search. *)
-Hint Extern 1 (ElemOf _ (listset _)) =>
- eapply @listset_elem_of : typeclass_instances.
-Hint Extern 1 (Empty (listset _)) =>
- eapply @listset_empty : typeclass_instances.
-Hint Extern 1 (Singleton _ (listset _)) =>
- eapply @listset_singleton : typeclass_instances.
-Hint Extern 1 (Union (listset _)) =>
- eapply @listset_union : typeclass_instances.
-Hint Extern 1 (Intersection (listset _)) =>
- eapply @listset_intersection : typeclass_instances.
-Hint Extern 1 (Difference (listset _)) =>
- eapply @listset_difference : typeclass_instances.
-Hint Extern 1 (Elements _ (listset _)) =>
- eapply @listset_elems : typeclass_instances.
-
Instance listset_ret: MRet listset := λ A x, {[ x ]}.
Instance listset_fmap: FMap listset := λ A B f l,
let (l') := l in Listset (f <$> l').
@@ -86,7 +70,7 @@ Instance listset_bind: MBind listset := λ A B f l,
let (l') := l in Listset (mbind (listset_car ∘ f) l').
Instance listset_join: MJoin listset := λ A, mbind id.
-Instance: CollectionMonad listset.
+Instance listset_collection_monad : CollectionMonad listset.
Proof.
split.
- intros. apply _.
diff --git a/theories/mapset.v b/theories/mapset.v
index bb44b6711ea679d48c9dc2828cb7253a11615167..bb7c479d8678dcdadcadaa54e83f1eafbf133148 100644
--- a/theories/mapset.v
+++ b/theories/mapset.v
@@ -14,18 +14,18 @@ Arguments mapset_car {_} _ : assert.
Section mapset.
Context `{FinMap K M}.
-Instance mapset_elem_of: ElemOf K (mapset M) := λ x X,
+Global Instance mapset_elem_of: ElemOf K (mapset M) := λ x X,
mapset_car X !! x = Some ().
-Instance mapset_empty: Empty (mapset M) := Mapset ∅.
-Instance mapset_singleton: Singleton K (mapset M) := λ x,
+Global Instance mapset_empty: Empty (mapset M) := Mapset ∅.
+Global Instance mapset_singleton: Singleton K (mapset M) := λ x,
Mapset {[ x := () ]}.
-Instance mapset_union: Union (mapset M) := λ X1 X2,
+Global Instance mapset_union: Union (mapset M) := λ X1 X2,
let (m1) := X1 in let (m2) := X2 in Mapset (m1 ∪ m2).
-Instance mapset_intersection: Intersection (mapset M) := λ X1 X2,
+Global Instance mapset_intersection: Intersection (mapset M) := λ X1 X2,
let (m1) := X1 in let (m2) := X2 in Mapset (m1 ∩ m2).
-Instance mapset_difference: Difference (mapset M) := λ X1 X2,
+Global Instance mapset_difference: Difference (mapset M) := λ X1 X2,
let (m1) := X1 in let (m2) := X2 in Mapset (m1 ∖ m2).
-Instance mapset_elems: Elements K (mapset M) := λ X,
+Global Instance mapset_elements: Elements K (mapset M) := λ X,
let (m) := X in (map_to_list m).*1.
Lemma mapset_eq (X1 X2 : mapset M) : X1 = X2 ↔ ∀ x, x ∈ X1 ↔ x ∈ X2.
@@ -35,7 +35,7 @@ Proof.
f_equal. apply map_eq. intros i. apply option_eq. intros []. by apply E.
Qed.
-Instance: Collection K (mapset M).
+Instance mapset_collection: Collection K (mapset M).
Proof.
split; [split | | ].
- unfold empty, elem_of, mapset_empty, mapset_elem_of.
@@ -54,17 +54,17 @@ Proof.
intros [m1] [m2] ?. simpl. rewrite lookup_difference_Some.
destruct (m2 !! x) as [[]|]; intuition congruence.
Qed.
-Global Instance: LeibnizEquiv (mapset M).
+Global Instance mapset_leibniz : LeibnizEquiv (mapset M).
Proof. intros ??. apply mapset_eq. Qed.
-Global Instance: FinCollection K (mapset M).
+Global Instance mapset_fin_collection : FinCollection K (mapset M).
Proof.
split.
- apply _.
- - unfold elements, elem_of at 2, mapset_elems, mapset_elem_of.
+ - unfold elements, elem_of at 2, mapset_elements, mapset_elem_of.
intros [m] x. simpl. rewrite elem_of_list_fmap. split.
+ intros ([y []] &?& Hy). subst. by rewrite <-elem_of_map_to_list.
+ intros. exists (x, ()). by rewrite elem_of_map_to_list.
- - unfold elements, mapset_elems. intros [m]. simpl.
+ - unfold elements, mapset_elements. intros [m]. simpl.
apply NoDup_fst_map_to_list.
Qed.
@@ -127,21 +127,4 @@ Proof.
Qed.
End mapset.
-(** These instances are declared using [Hint Extern] to avoid too
-eager type class search. *)
-Hint Extern 1 (ElemOf _ (mapset _)) =>
- eapply @mapset_elem_of : typeclass_instances.
-Hint Extern 1 (Empty (mapset _)) =>
- eapply @mapset_empty : typeclass_instances.
-Hint Extern 1 (Singleton _ (mapset _)) =>
- eapply @mapset_singleton : typeclass_instances.
-Hint Extern 1 (Union (mapset _)) =>
- eapply @mapset_union : typeclass_instances.
-Hint Extern 1 (Intersection (mapset _)) =>
- eapply @mapset_intersection : typeclass_instances.
-Hint Extern 1 (Difference (mapset _)) =>
- eapply @mapset_difference : typeclass_instances.
-Hint Extern 1 (Elements _ (mapset _)) =>
- eapply @mapset_elems : typeclass_instances.
-
Arguments mapset_eq_dec : simpl never.
diff --git a/theories/option.v b/theories/option.v
index 7a2c27f93058fa41508a092202182c643a68b5a6..9f5b7efb4b9bad18636185ae0adc230ec56f82a4 100644
--- a/theories/option.v
+++ b/theories/option.v
@@ -207,7 +207,7 @@ Proof. by destruct mx. Qed.
Lemma option_fmap_ext {A B} (f g : A → B) mx :
(∀ x, f x = g x) → f <$> mx = g <$> mx.
Proof. intros; destruct mx; f_equal/=; auto. Qed.
-Lemma option_fmap_equiv_ext `{Equiv A, Equiv B} (f g : A → B) mx :
+Lemma option_fmap_equiv_ext `{Equiv A, Equiv B} (f g : A → B) (mx : option A) :
(∀ x, f x ≡ g x) → f <$> mx ≡ g <$> mx.
Proof. destruct mx; constructor; auto. Qed.
Lemma option_fmap_bind {A B C} (f : A → B) (g : B → option C) mx :
@@ -306,13 +306,15 @@ Section union_intersection_difference.
Proof. by intros [?|]. Qed.
Global Instance union_with_right_id : RightId (=) None (union_with f).
Proof. by intros [?|]. Qed.
- Global Instance union_with_comm : Comm (=) f → Comm (=) (union_with f).
+ Global Instance union_with_comm :
+ Comm (=) f → Comm (=) (union_with (M:=option A) f).
Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm f). Qed.
Global Instance intersection_with_left_ab : LeftAbsorb (=) None (intersection_with f).
Proof. by intros [?|]. Qed.
Global Instance intersection_with_right_ab : RightAbsorb (=) None (intersection_with f).
Proof. by intros [?|]. Qed.
- Global Instance difference_with_comm : Comm (=) f → Comm (=) (intersection_with f).
+ Global Instance difference_with_comm :
+ Comm (=) f → Comm (=) (intersection_with (M:=option A) f).
Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm f). Qed.
Global Instance difference_with_right_id : RightId (=) None (difference_with f).
Proof. by intros [?|]. Qed.
diff --git a/theories/set.v b/theories/set.v
index 80f060ec6b7cda4b28cdb1cec45bc5202a6d8c00..9da4261094bb5cfa4d82e6bb0e0ca3f1a711a029 100644
--- a/theories/set.v
+++ b/theories/set.v
@@ -24,7 +24,7 @@ Instance set_difference {A} : Difference (set A) := λ X1 X2,
Instance set_collection : Collection A (set A).
Proof. split; [split | |]; by repeat intro. Qed.
-Lemma elem_of_top {A} (x : A) : x ∈ ⊤ ↔ True.
+Lemma elem_of_top {A} (x : A) : x ∈ (⊤ : set A) ↔ True.
Proof. done. Qed.
Lemma elem_of_mkSet {A} (P : A → Prop) x : x ∈ {[ x | P x ]} ↔ P x.
Proof. done. Qed.
@@ -40,7 +40,7 @@ Instance set_bind : MBind set := λ A B (f : A → set B) (X : set A),
Instance set_fmap : FMap set := λ A B (f : A → B) (X : set A),
{[ b | ∃ a, b = f a ∧ a ∈ X ]}.
Instance set_join : MJoin set := λ A (XX : set (set A)),
- {[ a | ∃ X, a ∈ X ∧ X ∈ XX ]}.
+ {[ a | ∃ X : set A, a ∈ X ∧ X ∈ XX ]}.
Instance set_collection_monad : CollectionMonad set.
Proof. by split; try apply _. Qed.