theories/coGset.v → stdpp/coGset.v

@@ -14,11 +14,11 @@ Set Default Proof Using "Type*".
 Inductive coGset `{Countable A} :=
   | FinGSet (X : gset A)
   | CoFinGset (X : gset A).
-Arguments coGset _ {_ _} : assert.
+Global Arguments coGset _ {_ _} : assert.
 
-Instance coGset_eq_dec `{Countable A} : EqDecision (coGset A).
+Global Instance coGset_eq_dec `{Countable A} : EqDecision (coGset A).
 Proof. solve_decision. Defined.
-Instance coGset_countable `{Countable A} : Countable (coGset A).
+Global Instance coGset_countable `{Countable A} : Countable (coGset A).
 Proof.
   apply (inj_countable'
     (λ X, match X with FinGSet X => inl X | CoFinGset X => inr X end)
@@ -80,7 +80,7 @@ Section coGset.
   Qed.
 End coGset.
 
-Instance coGset_elem_of_dec `{Countable A} : RelDecision (∈@{coGset A}) :=
+Global Instance coGset_elem_of_dec `{Countable A} : RelDecision (∈@{coGset A}) :=
   λ x X,
   match X with
   | FinGSet X => decide_rel elem_of x X
@@ -92,7 +92,7 @@ Section infinite.
 
   Global Instance coGset_leibniz : LeibnizEquiv (coGset A).
   Proof.
-    intros [X|X] [Y|Y]; rewrite elem_of_equiv;
+    intros [X|X] [Y|Y]; rewrite set_equiv;
     unfold elem_of, coGset_elem_of; simpl; intros HXY.
     - f_equal. by apply leibniz_equiv.
     - by destruct (exist_fresh (X ∪ Y)) as [? [? ?%HXY]%not_elem_of_union].
@@ -138,7 +138,7 @@ End infinite.
 Definition coGpick `{Countable A, Infinite A} (X : coGset A) : A :=
   fresh (match X with FinGSet _ => ∅ | CoFinGset X => X end).
 
-Lemma coGpick_elem_of `{Countable A, Infinite A} X :
+Lemma coGpick_elem_of `{Countable A, Infinite A} (X : coGset A) :
   ¬set_finite X → coGpick X ∈ X.
 Proof.
   unfold coGpick.
@@ -192,15 +192,5 @@ Lemma elem_of_coGset_to_top_set `{Countable A, TopSet A C} X x :
   x ∈@{C} coGset_to_top_set X ↔ x ∈ X.
 Proof. destruct X; set_solver. Qed.
 
-(** * Domain of finite maps *)
-Instance coGset_dom `{Countable K} {A} : Dom (gmap K A) (coGset K) := λ m,
-  gset_to_coGset (dom _ m).
-Instance coGset_dom_spec `{Countable K} : FinMapDom K (gmap K) (coGset K).
-Proof.
-  split; try apply _. intros B m i. unfold dom, coGset_dom.
-  by rewrite elem_of_gset_to_coGset, elem_of_dom.
-Qed.
-
-Typeclasses Opaque coGset_elem_of coGset_empty coGset_top coGset_singleton.
-Typeclasses Opaque coGset_union coGset_intersection coGset_difference.
-Typeclasses Opaque coGset_dom.
+Global Typeclasses Opaque coGset_elem_of coGset_empty coGset_top coGset_singleton.
+Global Typeclasses Opaque coGset_union coGset_intersection coGset_difference.

theories/coPset.v → stdpp/coPset.v

@@ -18,7 +18,7 @@ Local Open Scope positive_scope.
 Inductive coPset_raw :=
   | coPLeaf : bool → coPset_raw
   | coPNode : bool → coPset_raw → coPset_raw → coPset_raw.
-Instance coPset_raw_eq_dec : EqDecision coPset_raw.
+Global Instance coPset_raw_eq_dec : EqDecision coPset_raw.
 Proof. solve_decision. Defined.
 
 Fixpoint coPset_wf (t : coPset_raw) : bool :=
@@ -26,9 +26,16 @@ Fixpoint coPset_wf (t : coPset_raw) : bool :=
   | coPLeaf _ => true
   | coPNode true (coPLeaf true) (coPLeaf true) => false
   | coPNode false (coPLeaf false) (coPLeaf false) => false
-  | coPNode b l r => coPset_wf l && coPset_wf r
+  | coPNode _ l r => coPset_wf l && coPset_wf r
   end.
-Arguments coPset_wf !_ / : simpl nomatch, assert.
+Global Arguments coPset_wf !_ / : simpl nomatch, assert.
+
+Lemma coPNode_wf b l r :
+  coPset_wf l → coPset_wf r →
+  (l = coPLeaf true → r = coPLeaf true → b = true → False) →
+  (l = coPLeaf false → r = coPLeaf false → b = false → False) →
+  coPset_wf (coPNode b l r).
+Proof. destruct b, l as [[]|], r as [[]|]; naive_solver. Qed.
 
 Lemma coPNode_wf_l b l r : coPset_wf (coPNode b l r) → coPset_wf l.
 Proof. destruct b, l as [[]|],r as [[]|]; simpl; rewrite ?andb_True; tauto. Qed.
@@ -42,10 +49,10 @@ Definition coPNode' (b : bool) (l r : coPset_raw) : coPset_raw :=
   | false, coPLeaf false, coPLeaf false => coPLeaf false
   | _, _, _ => coPNode b l r
   end.
-Arguments coPNode' : simpl never.
-Lemma coPNode_wf b l r : coPset_wf l → coPset_wf r → coPset_wf (coPNode' b l r).
+Global Arguments coPNode' : simpl never.
+Lemma coPNode'_wf b l r : coPset_wf l → coPset_wf r → coPset_wf (coPNode' b l r).
 Proof. destruct b, l as [[]|], r as [[]|]; simpl; auto. Qed.
-Hint Resolve coPNode_wf : core.
+Global Hint Resolve coPNode'_wf : core.
 
 Fixpoint coPset_elem_of_raw (p : positive) (t : coPset_raw) {struct t} : bool :=
   match t, p with
@@ -55,7 +62,7 @@ Fixpoint coPset_elem_of_raw (p : positive) (t : coPset_raw) {struct t} : bool :=
   | coPNode _ _ r, p~1 => coPset_elem_of_raw p r
   end.
 Local Notation e_of := coPset_elem_of_raw.
-Arguments coPset_elem_of_raw _ !_ / : simpl nomatch, assert.
+Global Arguments coPset_elem_of_raw _ !_ / : simpl nomatch, assert.
 Lemma coPset_elem_of_node b l r p :
   e_of p (coPNode' b l r) = e_of p (coPNode b l r).
 Proof. by destruct p, b, l as [[]|], r as [[]|]. Qed.
@@ -87,7 +94,7 @@ Fixpoint coPset_singleton_raw (p : positive) : coPset_raw :=
   | p~0 => coPNode' false (coPset_singleton_raw p) (coPLeaf false)
   | p~1 => coPNode' false (coPLeaf false) (coPset_singleton_raw p)
   end.
-Instance coPset_union_raw : Union coPset_raw :=
+Global Instance coPset_union_raw : Union coPset_raw :=
   fix go t1 t2 := let _ : Union _ := @go in
   match t1, t2 with
   | coPLeaf false, coPLeaf false => coPLeaf false
@@ -98,7 +105,7 @@ Instance coPset_union_raw : Union coPset_raw :=
   | coPNode b1 l1 r1, coPNode b2 l2 r2 => coPNode' (b1||b2) (l1 ∪ l2) (r1 ∪ r2)
   end.
 Local Arguments union _ _!_ !_ / : assert.
-Instance coPset_intersection_raw : Intersection coPset_raw :=
+Global Instance coPset_intersection_raw : Intersection coPset_raw :=
   fix go t1 t2 := let _ : Intersection _ := @go in
   match t1, t2 with
   | coPLeaf true, coPLeaf true => coPLeaf true
@@ -152,22 +159,22 @@ Qed.
 (** * Packed together + set operations *)
 Definition coPset := { t | coPset_wf t }.
 
-Instance coPset_singleton : Singleton positive coPset := λ p,
+Global Instance coPset_singleton : Singleton positive coPset := λ p,
   coPset_singleton_raw p ↾ coPset_singleton_wf _.
-Instance coPset_elem_of : ElemOf positive coPset := λ p X, e_of p (`X).
-Instance coPset_empty : Empty coPset := coPLeaf false ↾ I.
-Instance coPset_top : Top coPset := coPLeaf true ↾ I.
-Instance coPset_union : Union coPset := λ X Y,
+Global Instance coPset_elem_of : ElemOf positive coPset := λ p X, e_of p (`X).
+Global Instance coPset_empty : Empty coPset := coPLeaf false ↾ I.
+Global Instance coPset_top : Top coPset := coPLeaf true ↾ I.
+Global Instance coPset_union : Union coPset := λ X Y,
   let (t1,Ht1) := X in let (t2,Ht2) := Y in
   (t1 ∪ t2) ↾ coPset_union_wf _ _ Ht1 Ht2.
-Instance coPset_intersection : Intersection coPset := λ X Y,
+Global Instance coPset_intersection : Intersection coPset := λ X Y,
   let (t1,Ht1) := X in let (t2,Ht2) := Y in
   (t1 ∩ t2) ↾ coPset_intersection_wf _ _ Ht1 Ht2.
-Instance coPset_difference : Difference coPset := λ X Y,
+Global Instance coPset_difference : Difference coPset := λ X Y,
   let (t1,Ht1) := X in let (t2,Ht2) := Y in
   (t1 ∩ coPset_opp_raw t2) ↾ coPset_intersection_wf _ _ Ht1 (coPset_opp_wf _).
 
-Instance coPset_top_set : TopSet positive coPset.
+Global Instance coPset_top_set : TopSet positive coPset.
 Proof.
   split; [split; [split| |]|].
   - by intros ??.
@@ -184,25 +191,25 @@ Qed.
 
 (** Iris and specifically [solve_ndisj] heavily rely on this hint. *)
 Local Definition coPset_top_subseteq := top_subseteq (C:=coPset).
-Hint Resolve coPset_top_subseteq : core.
+Global Hint Resolve coPset_top_subseteq : core.
 
-Instance coPset_leibniz : LeibnizEquiv coPset.
+Global Instance coPset_leibniz : LeibnizEquiv coPset.
 Proof.
-  intros X Y; rewrite elem_of_equiv; intros HXY.
+  intros X Y; rewrite set_equiv; intros HXY.
   apply (sig_eq_pi _), coPset_eq; try apply @proj2_sig.
   intros p; apply eq_bool_prop_intro, (HXY p).
 Qed.
 
-Instance coPset_elem_of_dec : RelDecision (∈@{coPset}).
+Global Instance coPset_elem_of_dec : RelDecision (∈@{coPset}).
 Proof. solve_decision. Defined.
-Instance coPset_equiv_dec : RelDecision (≡@{coPset}).
+Global Instance coPset_equiv_dec : RelDecision (≡@{coPset}).
 Proof. refine (λ X Y, cast_if (decide (X = Y))); abstract (by fold_leibniz). Defined.
-Instance mapset_disjoint_dec : RelDecision (##@{coPset}).
+Global Instance mapset_disjoint_dec : RelDecision (##@{coPset}).
 Proof.
  refine (λ X Y, cast_if (decide (X ∩ Y = ∅)));
   abstract (by rewrite disjoint_intersection_L).
 Defined.
-Instance mapset_subseteq_dec : RelDecision (⊆@{coPset}).
+Global Instance mapset_subseteq_dec : RelDecision (⊆@{coPset}).
 Proof.
  refine (λ X Y, cast_if (decide (X ∪ Y = Y))); abstract (by rewrite subseteq_union_L).
 Defined.
@@ -221,7 +228,7 @@ Proof.
   unfold set_finite, elem_of at 1, coPset_elem_of; simpl; clear Ht; split.
   - induction t as [b|b l IHl r IHr]; simpl.
     { destruct b; simpl; [intros [l Hl]|done].
-      by apply (is_fresh (list_to_set l : Pset)), elem_of_list_to_set, Hl. }
+      by apply (infinite_is_fresh l), Hl. }
     intros [ll Hll]; rewrite andb_True; split.
     + apply IHl; exists (omap (maybe (~0)) ll); intros i.
       rewrite elem_of_list_omap; intros; exists (i~0); auto.
@@ -232,14 +239,17 @@ Proof.
     exists ([1] ++ ((~0) <$> ll) ++ ((~1) <$> rl))%list; intros [i|i|]; simpl;
       rewrite elem_of_cons, elem_of_app, !elem_of_list_fmap; naive_solver.
 Qed.
-Instance coPset_finite_dec (X : coPset) : Decision (set_finite X).
+Global Instance coPset_finite_dec (X : coPset) : Decision (set_finite X).
 Proof.
   refine (cast_if (decide (coPset_finite (`X)))); by rewrite coPset_finite_spec.
 Defined.
 
 (** * Pick element from infinite sets *)
-(* Implemented using depth-first search, which results in very unbalanced
-trees. *)
+(* The function [coPpick X] gives an element that is in the set [X], provided
+that the set [X] is infinite. Note that [coPpick] function is implemented by
+depth-first search, so using it repeatedly to obtain elements [x], and
+inserting these elements [x] into the set [X], will give rise to a very
+unbalanced tree. *)
 Fixpoint coPpick_raw (t : coPset_raw) : option positive :=
   match t with
   | coPLeaf true | coPNode true _ _ => Some 1
@@ -269,89 +279,109 @@ Proof.
 Qed.
 
 (** * Conversion to psets *)
-Fixpoint coPset_to_Pset_raw (t : coPset_raw) : Pmap_raw () :=
+Fixpoint coPset_to_Pset_raw (t : coPset_raw) : Pmap () :=
   match t with
-  | coPLeaf _ => PLeaf
-  | coPNode false l r => PNode' None (coPset_to_Pset_raw l) (coPset_to_Pset_raw r)
-  | coPNode true l r => PNode (Some ()) (coPset_to_Pset_raw l) (coPset_to_Pset_raw r)
+  | coPLeaf _ => PEmpty
+  | coPNode false l r => pmap.PNode (coPset_to_Pset_raw l) None (coPset_to_Pset_raw r)
+  | coPNode true l r => pmap.PNode (coPset_to_Pset_raw l) (Some ()) (coPset_to_Pset_raw r)
   end.
-Lemma coPset_to_Pset_wf t : coPset_wf t → Pmap_wf (coPset_to_Pset_raw t).
-Proof. induction t as [|[]]; simpl; eauto using PNode_wf. Qed.
 Definition coPset_to_Pset (X : coPset) : Pset :=
-  let (t,Ht) := X in Mapset (PMap (coPset_to_Pset_raw t) (coPset_to_Pset_wf _ Ht)).
+  let (t,Ht) := X in Mapset (coPset_to_Pset_raw t).
 Lemma elem_of_coPset_to_Pset X i : set_finite X → i ∈ coPset_to_Pset X ↔ i ∈ X.
 Proof.
   rewrite coPset_finite_spec; destruct X as [t Ht].
   change (coPset_finite t → coPset_to_Pset_raw t !! i = Some () ↔ e_of i t).
   clear Ht; revert i; induction t as [[]|[] l IHl r IHr]; intros [i|i|];
-    simpl; rewrite ?andb_True, ?PNode_lookup; naive_solver.
+    simpl; rewrite ?andb_True, ?pmap.Pmap_lookup_PNode; naive_solver.
 Qed.
 
 (** * Conversion from psets *)
-Fixpoint Pset_to_coPset_raw (t : Pmap_raw ()) : coPset_raw :=
-  match t with
-  | PLeaf => coPLeaf false
-  | PNode None l r => coPNode false (Pset_to_coPset_raw l) (Pset_to_coPset_raw r)
-  | PNode (Some _) l r => coPNode true (Pset_to_coPset_raw l) (Pset_to_coPset_raw r)
-  end.
-Lemma Pset_to_coPset_wf t : Pmap_wf t → coPset_wf (Pset_to_coPset_raw t).
+Definition Pset_to_coPset_raw_aux (go : Pmap_ne () → coPset_raw)
+    (mt : Pmap ()) : coPset_raw :=
+  match mt with PNodes t => go t | PEmpty => coPLeaf false end.
+Fixpoint Pset_ne_to_coPset_raw (t : Pmap_ne ()) : coPset_raw :=
+  pmap.Pmap_ne_case t $ λ ml mx mr,
+    coPNode match mx with Some _ => true | None => false end
+      (Pset_to_coPset_raw_aux Pset_ne_to_coPset_raw ml)
+      (Pset_to_coPset_raw_aux Pset_ne_to_coPset_raw mr).
+Definition Pset_to_coPset_raw : Pmap () → coPset_raw :=
+  Pset_to_coPset_raw_aux Pset_ne_to_coPset_raw.
+
+Lemma Pset_to_coPset_raw_PNode ml mx mr :
+  pmap.PNode_valid ml mx mr →
+  Pset_to_coPset_raw (pmap.PNode ml mx mr) =
+    coPNode match mx with Some _ => true | None => false end
+    (Pset_to_coPset_raw ml) (Pset_to_coPset_raw mr).
+Proof. by destruct ml, mx, mr. Qed.
+Lemma Pset_to_coPset_raw_wf t : coPset_wf (Pset_to_coPset_raw t).
 Proof.
-  induction t as [|[] l IHl r IHr]; simpl; rewrite ?andb_True; auto.
-  - intros [??]; destruct l as [|[]], r as [|[]]; simpl in *; auto.
-  - destruct l as [|[]], r as [|[]]; simpl in *; rewrite ?andb_true_r;
-      rewrite ?andb_True; rewrite ?andb_True in IHl, IHr; intuition.
+  induction t as [|ml mx mr] using pmap.Pmap_ind; [done|].
+  rewrite Pset_to_coPset_raw_PNode by done.
+  apply coPNode_wf; [done|done|..];
+    destruct mx; destruct ml using pmap.Pmap_ind; destruct mr using pmap.Pmap_ind;
+    rewrite ?Pset_to_coPset_raw_PNode by done; naive_solver.
 Qed.
 Lemma elem_of_Pset_to_coPset_raw i t : e_of i (Pset_to_coPset_raw t) ↔ t !! i = Some ().
-Proof. by revert i; induction t as [|[[]|]]; intros []; simpl; auto; split. Qed.
+Proof.
+  revert i. induction t as [|ml mx mr] using pmap.Pmap_ind; [done|].
+  intros []; rewrite Pset_to_coPset_raw_PNode,
+    pmap.Pmap_lookup_PNode by done; destruct mx as [[]|]; naive_solver.
+Qed.
 Lemma Pset_to_coPset_raw_finite t : coPset_finite (Pset_to_coPset_raw t).
-Proof. induction t as [|[[]|]]; simpl; rewrite ?andb_True; auto. Qed.
+Proof.
+  induction t as [|ml mx mr] using pmap.Pmap_ind; [done|].
+  rewrite Pset_to_coPset_raw_PNode by done. destruct mx; naive_solver.
+Qed.
 
 Definition Pset_to_coPset (X : Pset) : coPset :=
-  let 'Mapset (PMap t Ht) := X in Pset_to_coPset_raw t ↾ Pset_to_coPset_wf _ Ht.
+  let 'Mapset t := X in Pset_to_coPset_raw t ↾ Pset_to_coPset_raw_wf _.
 Lemma elem_of_Pset_to_coPset X i : i ∈ Pset_to_coPset X ↔ i ∈ X.
-Proof. destruct X as [[t ?]]; apply elem_of_Pset_to_coPset_raw. Qed.
+Proof. destruct X; apply elem_of_Pset_to_coPset_raw. Qed.
 Lemma Pset_to_coPset_finite X : set_finite (Pset_to_coPset X).
-Proof.
-  apply coPset_finite_spec; destruct X as [[t ?]]; apply Pset_to_coPset_raw_finite.
-Qed.
+Proof. apply coPset_finite_spec; destruct X; apply Pset_to_coPset_raw_finite. Qed.
 
 (** * Conversion to and from gsets of positives *)
-Lemma coPset_to_gset_wf (m : Pmap ()) : gmap_wf positive m.
-Proof. unfold gmap_wf. by rewrite bool_decide_spec. Qed.
 Definition coPset_to_gset (X : coPset) : gset positive :=
   let 'Mapset m := coPset_to_Pset X in
-  Mapset (GMap m (coPset_to_gset_wf m)).
+  Mapset (pmap_to_gmap m).
 
 Definition gset_to_coPset (X : gset positive) : coPset :=
-  let 'Mapset (GMap (PMap t Ht) _) := X in Pset_to_coPset_raw t ↾ Pset_to_coPset_wf _ Ht.
+  let 'Mapset m := X in
+  Pset_to_coPset_raw (gmap_to_pmap m) ↾ Pset_to_coPset_raw_wf _.
 
 Lemma elem_of_coPset_to_gset X i : set_finite X → i ∈ coPset_to_gset X ↔ i ∈ X.
 Proof.
-  intros ?. rewrite <-elem_of_coPset_to_Pset by done.
-  unfold coPset_to_gset. by destruct (coPset_to_Pset X).
+  intros ?. rewrite <-elem_of_coPset_to_Pset by done. destruct X as [X ?].
+  unfold elem_of, gset_elem_of, mapset_elem_of, coPset_to_gset; simpl.
+  by rewrite lookup_pmap_to_gmap.
 Qed.
 
 Lemma elem_of_gset_to_coPset X i : i ∈ gset_to_coPset X ↔ i ∈ X.
-Proof. destruct X as [[[t ?]]]; apply elem_of_Pset_to_coPset_raw. Qed.
+Proof.
+  destruct X as [m]. unfold elem_of, coPset_elem_of; simpl.
+  by rewrite elem_of_Pset_to_coPset_raw, lookup_gmap_to_pmap.
+Qed.
 Lemma gset_to_coPset_finite X : set_finite (gset_to_coPset X).
 Proof.
-  apply coPset_finite_spec; destruct X as [[[t ?]]]; apply Pset_to_coPset_raw_finite.
+  apply coPset_finite_spec; destruct X as [[?]]; apply Pset_to_coPset_raw_finite.
 Qed.
 
-(** * Domain of finite maps *)
-Instance Pmap_dom_coPset {A} : Dom (Pmap A) coPset := λ m, Pset_to_coPset (dom _ m).
-Instance Pmap_dom_coPset_spec: FinMapDom positive Pmap coPset.
+(** * Infinite sets *)
+Lemma coPset_infinite_finite (X : coPset) : set_infinite X ↔ ¬set_finite X.
 Proof.
-  split; try apply _; intros A m i; unfold dom, Pmap_dom_coPset.
-  by rewrite elem_of_Pset_to_coPset, elem_of_dom.
+  split; [intros ??; by apply (set_not_infinite_finite X)|].
+  intros Hfin xs. exists (coPpick (X ∖ list_to_set xs)).
+  cut (coPpick (X ∖ list_to_set xs) ∈ X ∖ list_to_set xs); [set_solver|].
+  apply coPpick_elem_of; intros Hfin'.
+  apply Hfin, (difference_finite_inv _ (list_to_set xs)), Hfin'.
+  apply list_to_set_finite.
 Qed.
-Instance gmap_dom_coPset {A} : Dom (gmap positive A) coPset := λ m,
-  gset_to_coPset (dom _ m).
-Instance gmap_dom_coPset_spec: FinMapDom positive (gmap positive) coPset.
+Lemma coPset_finite_infinite (X : coPset) : set_finite X ↔ ¬set_infinite X.
+Proof. rewrite coPset_infinite_finite. split; [tauto|apply dec_stable]. Qed.
+Global Instance coPset_infinite_dec (X : coPset) : Decision (set_infinite X).
 Proof.
-  split; try apply _; intros A m i; unfold dom, gmap_dom_coPset.
-  by rewrite elem_of_gset_to_coPset, elem_of_dom.
-Qed.
+  refine (cast_if (decide (¬set_finite X))); by rewrite coPset_infinite_finite.
+Defined.
 
 (** * Suffix sets *)
 Fixpoint coPset_suffixes_raw (p : positive) : coPset_raw :=
@@ -410,7 +440,7 @@ Proof.
 Qed.
 Lemma coPset_lr_union X : coPset_l X ∪ coPset_r X = X.
 Proof.
-  apply elem_of_equiv_L; intros p; apply eq_bool_prop_elim.
+  apply set_eq; intros p; apply eq_bool_prop_elim.
   destruct X as [t Ht]; simpl; clear Ht; rewrite coPset_elem_of_union.
   revert p; induction t as [[]|[]]; intros [?|?|]; simpl;
     rewrite ?coPset_elem_of_node; simpl;
@@ -433,3 +463,10 @@ Proof.
   exists (coPset_l X), (coPset_r X); eauto 10 using coPset_lr_union,
     coPset_lr_disjoint, coPset_l_finite, coPset_r_finite.
 Qed.
+Lemma coPset_split_infinite (X : coPset) :
+  set_infinite X →
+  ∃ X1 X2, X = X1 ∪ X2 ∧ X1 ∩ X2 = ∅ ∧ set_infinite X1 ∧ set_infinite X2.
+Proof.
+  setoid_rewrite coPset_infinite_finite.
+  eapply coPset_split.
+Qed.

theories/countable.v → stdpp/countable.v

 From Coq.QArith Require Import QArith_base Qcanon.
 From stdpp Require Export list numbers list_numbers fin.
+From stdpp Require Import well_founded.
 From stdpp Require Import options.
 Local Open Scope positive.
 
+(** Note that [Countable A] gives rise to [EqDecision A] by checking equality of
+the results of [encode]. This instance of [EqDecision A] is very inefficient, so
+the native decider is typically preferred for actual computation. To avoid
+overlapping instances, we include [EqDecision A] explicitly as a parameter of
+[Countable A]. *)
 Class Countable A `{EqDecision A} := {
   encode : A → positive;
   decode : positive → option A;
   decode_encode x : decode (encode x) = Some x
 }.
-Hint Mode Countable ! - : typeclass_instances.
-Arguments encode : simpl never.
-Arguments decode : simpl never.
+Global Hint Mode Countable ! - : typeclass_instances.
+Global Arguments encode : simpl never.
+Global Arguments decode : simpl never.
 
-Instance encode_inj `{Countable A} : Inj (=) (=) (encode (A:=A)).
+Global Instance encode_inj `{Countable A} : Inj (=) (=) (encode (A:=A)).
 Proof.
   intros x y Hxy; apply (inj Some).
   by rewrite <-(decode_encode x), Hxy, decode_encode.
@@ -22,7 +28,7 @@ Definition encode_nat `{Countable A} (x : A) : nat :=
   pred (Pos.to_nat (encode x)).
 Definition decode_nat `{Countable A} (i : nat) : option A :=
   decode (Pos.of_nat (S i)).
-Instance encode_nat_inj `{Countable A} : Inj (=) (=) (encode_nat (A:=A)).
+Global Instance encode_nat_inj `{Countable A} : Inj (=) (=) (encode_nat (A:=A)).
 Proof. unfold encode_nat; intros x y Hxy; apply (inj encode); lia. Qed.
 Lemma decode_encode_nat `{Countable A} (x : A) : decode_nat (encode_nat x) = Some x.
 Proof.
@@ -35,7 +41,7 @@ Definition encode_Z `{Countable A} (x : A) : Z :=
   Zpos (encode x).
 Definition decode_Z `{Countable A} (i : Z) : option A :=
   match i with Zpos i => decode i | _ => None end.
-Instance encode_Z_inj `{Countable A} : Inj (=) (=) (encode_Z (A:=A)).
+Global Instance encode_Z_inj `{Countable A} : Inj (=) (=) (encode_Z (A:=A)).
 Proof. unfold encode_Z; intros x y Hxy; apply (inj encode); lia. Qed.
 Lemma decode_encode_Z `{Countable A} (x : A) : decode_Z (encode_Z x) = Some x.
 Proof. apply decode_encode. Qed.
@@ -55,10 +61,9 @@ Section choice.
     { intros help. by apply (help (encode x)). }
     intros i. induction i as [|i IH] using Pos.peano_ind; intros p ??.
     { constructor. intros j. assert (p = encode x) by lia; subst.
-      inversion 1 as [? Hd|?? Hd]; subst;
-        rewrite decode_encode in Hd; congruence. }
+      inv 1 as [? Hd|?? Hd]; rewrite decode_encode in Hd; congruence. }
     constructor. intros j.
-    inversion 1 as [? Hd|? y Hd]; subst; auto with lia.
+    inv 1 as [? Hd|? y Hd]; auto with lia.
   Qed.
 
   Context `{∀ x, Decision (P x)}.
@@ -84,6 +89,25 @@ Section choice.
   Definition choice (HA : ∃ x, P x) : { x | P x } := _↾choose_correct HA.
 End choice.
 
+Section choice_proper.
+  Context `{Countable A}.
+  Context (P1 P2 : A → Prop) `{∀ x, Decision (P1 x)} `{∀ x, Decision (P2 x)}.
+  Context (Heq : ∀ x, P1 x ↔ P2 x).
+
+  Lemma choose_go_proper {i} (acc1 acc2 : Acc (choose_step _) i) :
+    choose_go P1 acc1 = choose_go P2 acc2.
+  Proof using Heq.
+    induction acc1 as [i a1 IH] using Acc_dep_ind;
+      destruct acc2 as [acc2]; simpl.
+    destruct (Some_dec _) as [[x Hx]|]; [|done].
+    do 2 case_decide; done || exfalso; naive_solver.
+  Qed.
+
+  Lemma choose_proper p1 p2 :
+    choose P1 p1 = choose P2 p2.
+  Proof using Heq. apply choose_go_proper. Qed.
+End choice_proper.
+
 Lemma surj_cancel `{Countable A} `{EqDecision B}
   (f : A → B) `{!Surj (=) f} : { g : B → A & Cancel (=) f g }.
 Proof.
@@ -97,7 +121,7 @@ Section inj_countable.
   Context `{Countable A, EqDecision B}.
   Context (f : B → A) (g : A → option B) (fg : ∀ x, g (f x) = Some x).
 
-  Program Instance inj_countable : Countable B :=
+  Program Definition inj_countable : Countable B :=
     {| encode y := encode (f y); decode p := x ← decode p; g x |}.
   Next Obligation. intros y; simpl; rewrite decode_encode; eauto. Qed.
 End inj_countable.
@@ -106,29 +130,29 @@ Section inj_countable'.
   Context `{Countable A, EqDecision B}.
   Context (f : B → A) (g : A → B) (fg : ∀ x, g (f x) = x).
 
-  Program Instance inj_countable' : Countable B := inj_countable f (Some ∘ g) _.
+  Program Definition inj_countable' : Countable B := inj_countable f (Some ∘ g) _.
   Next Obligation. intros x. by f_equal/=. Qed.
 End inj_countable'.
 
 (** ** Empty *)
-Program Instance Empty_set_countable : Countable Empty_set :=
+Global Program Instance Empty_set_countable : Countable Empty_set :=
   {| encode u := 1; decode p := None |}.
 Next Obligation. by intros []. Qed.
 
 (** ** Unit *)
-Program Instance unit_countable : Countable unit :=
+Global Program Instance unit_countable : Countable unit :=
   {| encode u := 1; decode p := Some () |}.
 Next Obligation. by intros []. Qed.
 
 (** ** Bool *)
-Program Instance bool_countable : Countable bool := {|
+Global Program Instance bool_countable : Countable bool := {|
   encode b := if b then 1 else 2;
   decode p := Some match p return bool with 1 => true | _ => false end
 |}.
 Next Obligation. by intros []. Qed.
 
 (** ** Option *)
-Program Instance option_countable `{Countable A} : Countable (option A) := {|
+Global Program Instance option_countable `{Countable A} : Countable (option A) := {|
   encode o := match o with None => 1 | Some x => Pos.succ (encode x) end;
   decode p := if decide (p = 1) then Some None else Some <$> decode (Pos.pred p)
 |}.
@@ -138,7 +162,7 @@ Next Obligation.
 Qed.
 
 (** ** Sums *)
-Program Instance sum_countable `{Countable A} `{Countable B} :
+Global Program Instance sum_countable `{Countable A} `{Countable B} :
   Countable (A + B)%type := {|
     encode xy :=
       match xy with inl x => (encode x)~0 | inr y => (encode y)~1 end;
@@ -211,7 +235,7 @@ Proof.
   { intros p'. by induction p'; simplify_option_eq. }
   revert q. by induction p; intros [?|?|]; simplify_option_eq.
 Qed.
-Program Instance prod_countable `{Countable A} `{Countable B} :
+Global Program Instance prod_countable `{Countable A} `{Countable B} :
   Countable (A * B)%type := {|
     encode xy := prod_encode (encode (xy.1)) (encode (xy.2));
     decode p :=
@@ -238,9 +262,9 @@ Next Obligation.
 Qed.
 
 (** ** Numbers *)
-Instance pos_countable : Countable positive :=
+Global Instance pos_countable : Countable positive :=
   {| encode := id; decode := Some; decode_encode x := eq_refl |}.
-Program Instance N_countable : Countable N := {|
+Global Program Instance N_countable : Countable N := {|
   encode x := match x with N0 => 1 | Npos p => Pos.succ p end;
   decode p := if decide (p = 1) then Some 0%N else Some (Npos (Pos.pred p))
 |}.
@@ -248,18 +272,18 @@ Next Obligation.
   intros [|p]; simpl; [done|].
   by rewrite decide_False, Pos.pred_succ by (by destruct p).
 Qed.
-Program Instance Z_countable : Countable Z := {|
+Global Program Instance Z_countable : Countable Z := {|
   encode x := match x with Z0 => 1 | Zpos p => p~0 | Zneg p => p~1 end;
   decode p := Some match p with 1 => Z0 | p~0 => Zpos p | p~1 => Zneg p end
 |}.
 Next Obligation. by intros [|p|p]. Qed.
-Program Instance nat_countable : Countable nat :=
+Global Program Instance nat_countable : Countable nat :=
   {| encode x := encode (N.of_nat x); decode p := N.to_nat <$> decode p |}.
 Next Obligation.
   by intros x; lazy beta; rewrite decode_encode; csimpl; rewrite Nat2N.id.
 Qed.
 
-Program Instance Qc_countable : Countable Qc :=
+Global Program Instance Qc_countable : Countable Qc :=
   inj_countable
     (λ p : Qc, let 'Qcmake (x # y) _ := p return _ in (x,y))
     (λ q : Z * positive, let '(x,y) := q return _ in Some (Q2Qc (x # y))) _.
@@ -267,19 +291,19 @@ Next Obligation.
   intros [[x y] Hcan]. f_equal. apply Qc_is_canon. simpl. by rewrite Hcan.
 Qed.
 
-Program Instance Qp_countable : Countable Qp :=
+Global Program Instance Qp_countable : Countable Qp :=
   inj_countable
-    Qp_car
-    (λ p : Qc, guard (0 < p)%Qc as Hp; Some (mk_Qp p Hp)) _.
+    Qp_to_Qc
+    (λ p : Qc, Hp ← guard (0 < p)%Qc; Some (mk_Qp p Hp)) _.
 Next Obligation.
-  intros [p Hp]. unfold mguard, option_guard; simpl.
-  case_match; [|done]. f_equal. by apply Qp_eq.
+  intros [p Hp]. case_guard; simplify_eq/=; [|done].
+  f_equal. by apply Qp.to_Qc_inj_iff.
 Qed.
 
-Program Instance fin_countable n : Countable (fin n) :=
+Global Program Instance fin_countable n : Countable (fin n) :=
   inj_countable
     fin_to_nat
-    (λ m : nat, guard (m < n)%nat as Hm; Some (nat_to_fin Hm)) _.
+    (λ m : nat, Hm ← guard (m < n)%nat; Some (nat_to_fin Hm)) _.
 Next Obligation.
   intros n i; simplify_option_eq.
   - by rewrite nat_to_fin_to_nat.
@@ -289,13 +313,25 @@ Qed.
 (** ** Generic trees *)
 Local Close Scope positive.
 
+(** This type can help you construct a [Countable] instance for an arbitrary
+(even recursive) inductive datatype. The idea is tht you make [T] something like
+[T1 + T2 + ...], covering all the data types that can be contained inside your
+type.
+- Each non-recursive constructor to a [GenLeaf]. Different constructors must use
+  different variants of [T] to ensure they remain distinguishable!
+- Each recursive constructor to a [GenNode] where the [nat] is a (typically
+  small) constant representing the constructor itself, and then all the data in
+  the constructor (recursive or otherwise) is put into child nodes.
+
+This data type is the same as [GenTree.tree] in mathcomp, see
+https://github.com/math-comp/math-comp/blob/master/ssreflect/choice.v *)
 Inductive gen_tree (T : Type) : Type :=
   | GenLeaf : T → gen_tree T
   | GenNode : nat → list (gen_tree T) → gen_tree T.
-Arguments GenLeaf {_} _ : assert.
-Arguments GenNode {_} _ _ : assert.
+Global Arguments GenLeaf {_} _ : assert.
+Global Arguments GenNode {_} _ _ : assert.
 
-Instance gen_tree_dec `{EqDecision T} : EqDecision (gen_tree T).
+Global Instance gen_tree_dec `{EqDecision T} : EqDecision (gen_tree T).
 Proof.
  refine (
   fix go t1 t2 := let _ : EqDecision _ := @go in
@@ -330,13 +366,22 @@ Proof.
   - rewrite <-(assoc_L _). revert k. generalize ([inl (length ts, n)] ++ l).
     induction ts as [|t ts'' IH]; intros k ts'''; csimpl; auto.
     rewrite reverse_cons, <-!(assoc_L _), FIX; simpl; auto.
-  - simpl. by rewrite take_app_alt, drop_app_alt, reverse_involutive
-      by (by rewrite reverse_length).
+  - simpl. by rewrite take_app_length', drop_app_length', reverse_involutive
+      by (by rewrite length_reverse).
 Qed.
 
-Program Instance gen_tree_countable `{Countable T} : Countable (gen_tree T) :=
+Global Program Instance gen_tree_countable `{Countable T} : Countable (gen_tree T) :=
   inj_countable gen_tree_to_list (gen_tree_of_list []) _.
 Next Obligation.
   intros T ?? t.
   by rewrite <-(right_id_L [] _ (gen_tree_to_list _)), gen_tree_of_to_list.
 Qed.
+
+(** ** Sigma *)
+Global Program Instance countable_sig `{Countable A} (P : A → Prop)
+        `{!∀ x, Decision (P x), !∀ x, ProofIrrel (P x)} :
+  Countable { x : A | P x } :=
+  inj_countable proj1_sig (λ x, Hx ← guard (P x); Some (x ↾ Hx)) _.
+Next Obligation.
+  intros A ?? P ?? [x Hx]. by erewrite (option_guard_True_pi (P x)).
+Qed.

theories/decidable.v → stdpp/decidable.v

@@ -12,7 +12,7 @@ Proof. firstorder. Qed.
 
 Lemma Is_true_reflect (b : bool) : reflect b b.
 Proof. destruct b; [left; constructor | right; intros []]. Qed.
-Instance: Inj (=) (↔) Is_true.
+Global Instance: Inj (=) (↔) Is_true.
 Proof. intros [] []; simpl; intuition. Qed.
 
 Lemma decide_True {A} `{Decision P} (x y : A) :
@@ -21,13 +21,13 @@ Proof. destruct (decide P); tauto. Qed.
 Lemma decide_False {A} `{Decision P} (x y : A) :
   ¬P → (if decide P then x else y) = y.
 Proof. destruct (decide P); tauto. Qed.
-Lemma decide_iff {A} P Q `{Decision P, Decision Q} (x y : A) :
+Lemma decide_ext {A} P Q `{Decision P, Decision Q} (x y : A) :
   (P ↔ Q) → (if decide P then x else y) = (if decide Q then x else y).
 Proof. intros [??]. destruct (decide P), (decide Q); tauto. Qed.
 
-Lemma decide_left `{Decision P, !ProofIrrel P} (HP : P) : decide P = left HP.
+Lemma decide_True_pi `{Decision P, !ProofIrrel P} (HP : P) : decide P = left HP.
 Proof. destruct (decide P); [|contradiction]. f_equal. apply proof_irrel. Qed.
-Lemma decide_right `{Decision P, !ProofIrrel (¬ P)} (HP : ¬ P) : decide P = right HP.
+Lemma decide_False_pi `{Decision P, !ProofIrrel (¬ P)} (HP : ¬ P) : decide P = right HP.
 Proof. destruct (decide P); [contradiction|]. f_equal. apply proof_irrel. Qed.
 
 (** The tactic [destruct_decide] destructs a sumbool [dec]. If one of the
@@ -85,6 +85,79 @@ Notation cast_if_or3 S1 S2 S3 := (if S1 then left _ else cast_if_or S2 S3).
 Notation cast_if_not_or S1 S2 := (if S1 then cast_if S2 else left _).
 Notation cast_if_not S := (if S then right _ else left _).
 
+(** * Instances of [Decision] *)
+(** Instances of [Decision] for operators of propositional logic. *)
+(** The instances for [True] and [False] have a very high cost. If they are
+applied too eagerly, HO-unification could wrongfully instantiate TC instances
+with [λ .., True] or [λ .., False].
+See https://gitlab.mpi-sws.org/iris/stdpp/-/issues/165 *)
+Global Instance True_dec: Decision True | 1000 := left I.
+Global Instance False_dec: Decision False | 1000 := right (False_rect False).
+
+Global Instance Is_true_dec b : Decision (Is_true b).
+Proof. destruct b; simpl; apply _. Defined.
+
+Section prop_dec.
+  Context `(P_dec : Decision P) `(Q_dec : Decision Q).
+
+  Global Instance not_dec: Decision (¬P).
+  Proof. refine (cast_if_not P_dec); intuition. Defined.
+  Global Instance and_dec: Decision (P ∧ Q).
+  Proof. refine (cast_if_and P_dec Q_dec); intuition. Defined.
+  Global Instance or_dec: Decision (P ∨ Q).
+  Proof. refine (cast_if_or P_dec Q_dec); intuition. Defined.
+  Global Instance impl_dec: Decision (P → Q).
+  Proof. refine (if P_dec then cast_if Q_dec else left _); intuition. Defined.
+End prop_dec.
+Global Instance iff_dec `(P_dec : Decision P) `(Q_dec : Decision Q) :
+  Decision (P ↔ Q) := and_dec _ _.
+
+(** Instances of [Decision] for common data types. *)
+Global Instance bool_eq_dec : EqDecision bool.
+Proof. solve_decision. Defined.
+Global Instance unit_eq_dec : EqDecision unit.
+Proof. solve_decision. Defined.
+Global Instance Empty_set_eq_dec : EqDecision Empty_set.
+Proof. solve_decision. Defined.
+Global Instance prod_eq_dec `{EqDecision A, EqDecision B} : EqDecision (A * B).
+Proof. solve_decision. Defined.
+Global Instance sum_eq_dec `{EqDecision A, EqDecision B} : EqDecision (A + B).
+Proof. solve_decision. Defined.
+
+Global Instance uncurry_dec `(P_dec : ∀ (x : A) (y : B), Decision (P x y)) p :
+    Decision (uncurry P p) :=
+  match p as p return Decision (uncurry P p) with
+  | (x,y) => P_dec x y
+  end.
+
+Global Instance sig_eq_dec `(P : A → Prop) `{∀ x, ProofIrrel (P x), EqDecision A} :
+  EqDecision (sig P).
+Proof.
+ refine (λ x y, cast_if (decide (`x = `y))); rewrite sig_eq_pi; trivial.
+Defined.
+
+(** Some laws for decidable propositions *)
+Lemma not_and_l {P Q : Prop} `{Decision P} : ¬(P ∧ Q) ↔ ¬P ∨ ¬Q.
+Proof. destruct (decide P); tauto. Qed.
+Lemma not_and_r {P Q : Prop} `{Decision Q} : ¬(P ∧ Q) ↔ ¬P ∨ ¬Q.
+Proof. destruct (decide Q); tauto. Qed.
+Lemma not_and_l_alt {P Q : Prop} `{Decision P} : ¬(P ∧ Q) ↔ ¬P ∨ (¬Q ∧ P).
+Proof. destruct (decide P); tauto. Qed.
+Lemma not_and_r_alt {P Q : Prop} `{Decision Q} : ¬(P ∧ Q) ↔ (¬P ∧ Q) ∨ ¬Q.
+Proof. destruct (decide Q); tauto. Qed.
+
+Program Definition inj_eq_dec `{EqDecision A} {B} (f : B → A)
+  `{!Inj (=) (=) f} : EqDecision B := λ x y, cast_if (decide (f x = f y)).
+Solve Obligations with firstorder congruence.
+
+(** * Instances of [RelDecision] *)
+Definition flip_dec {A} (R : relation A) `{!RelDecision R} :
+  RelDecision (flip R) := λ x y, decide_rel R y x.
+(** We do not declare this as an actual instance since Coq can unify [flip ?R]
+with any relation. Coq's standard library is carrying out the same approach for
+the [Reflexive], [Transitive], etc, instance of [flip]. *)
+Global Hint Extern 3 (RelDecision (flip _)) => apply flip_dec : typeclass_instances.
+
 (** We can convert decidable propositions to booleans. *)
 Definition bool_decide (P : Prop) {dec : Decision P} : bool :=
   if dec then true else false.
@@ -99,7 +172,7 @@ Lemma decide_bool_decide P {Hdec: Decision P} {X : Type} (x1 x2 : X):
   (if decide P then x1 else x2) = (if bool_decide P then x1 else x2).
 Proof. unfold bool_decide, decide. destruct Hdec; reflexivity. Qed.
 
-Tactic Notation "case_bool_decide" "as" ident (Hd) :=
+Tactic Notation "case_bool_decide" "as" ident(Hd) :=
   match goal with
   | H : context [@bool_decide ?P ?dec] |- _ =>
     destruct_decide (@bool_decide_reflect P dec) as Hd
@@ -115,15 +188,15 @@ Lemma bool_decide_unpack (P : Prop) {dec : Decision P} : bool_decide P → P.
 Proof. rewrite bool_decide_spec; trivial. Qed.
 Lemma bool_decide_pack (P : Prop) {dec : Decision P} : P → bool_decide P.
 Proof. rewrite bool_decide_spec; trivial. Qed.
-Hint Resolve bool_decide_pack : core.
+Global Hint Resolve bool_decide_pack : core.
 
 Lemma bool_decide_eq_true (P : Prop) `{Decision P} : bool_decide P = true ↔ P.
 Proof. case_bool_decide; intuition discriminate. Qed.
 Lemma bool_decide_eq_false (P : Prop) `{Decision P} : bool_decide P = false ↔ ¬P.
 Proof. case_bool_decide; intuition discriminate. Qed.
-Lemma bool_decide_iff (P Q : Prop) `{Decision P, Decision Q} :
+Lemma bool_decide_ext (P Q : Prop) `{Decision P, Decision Q} :
   (P ↔ Q) → bool_decide P = bool_decide Q.
-Proof. repeat case_bool_decide; tauto. Qed.
+Proof. apply decide_ext. Qed.
 
 Lemma bool_decide_eq_true_1 P `{!Decision P}: bool_decide P = true → P.
 Proof. apply bool_decide_eq_true. Qed.
@@ -135,6 +208,40 @@ Proof. apply bool_decide_eq_false. Qed.
 Lemma bool_decide_eq_false_2 P `{!Decision P}: ¬P → bool_decide P = false.
 Proof. apply bool_decide_eq_false. Qed.
 
+Lemma bool_decide_True : bool_decide True = true.
+Proof. reflexivity. Qed.
+Lemma bool_decide_False : bool_decide False = false.
+Proof. reflexivity. Qed.
+Lemma bool_decide_not P `{Decision P} :
+  bool_decide (¬ P) = negb (bool_decide P).
+Proof. repeat case_bool_decide; intuition. Qed.
+Lemma bool_decide_or P Q `{Decision P, Decision Q} :
+  bool_decide (P ∨ Q) = bool_decide P || bool_decide Q.
+Proof. repeat case_bool_decide; intuition. Qed.
+Lemma bool_decide_and P Q `{Decision P, Decision Q} :
+  bool_decide (P ∧ Q) = bool_decide P && bool_decide Q.
+Proof. repeat case_bool_decide; intuition. Qed.
+Lemma bool_decide_impl P Q `{Decision P, Decision Q} :
+  bool_decide (P → Q) = implb (bool_decide P) (bool_decide Q).
+Proof. repeat case_bool_decide; intuition. Qed.
+Lemma bool_decide_iff P Q `{Decision P, Decision Q} :
+  bool_decide (P ↔ Q) = eqb (bool_decide P) (bool_decide Q).
+Proof. repeat case_bool_decide; intuition. Qed.
+
+(** The tactic [compute_done] solves the following kinds of goals:
+- Goals [P] where [Decidable P] can be derived.
+- Goals that compute to [True] or [x = x].
+
+The goal must be a ground term for this, i.e., not contain variables (that do
+not compute away). The goal is solved by using [vm_compute] and then using a
+trivial proof term ([I]/[eq_refl]). *)
+Tactic Notation "compute_done" :=
+  try apply (bool_decide_unpack _);
+  vm_compute;
+  first [ exact I | exact eq_refl ].
+Tactic Notation "compute_by" tactic(tac) :=
+  tac; compute_done.
+
 (** Backwards compatibility notations. *)
 Notation bool_decide_true := bool_decide_eq_true_2.
 Notation bool_decide_false := bool_decide_eq_false_2.
@@ -157,71 +264,3 @@ Proof. apply (sig_eq_pi _). Qed.
 Lemma dexists_proj1 `(P : A → Prop) `{∀ x, Decision (P x)} (x : dsig P) p :
   dexist (`x) p = x.
 Proof. apply dsig_eq; reflexivity. Qed.
-
-(** * Instances of [Decision] *)
-(** Instances of [Decision] for operators of propositional logic. *)
-Instance True_dec: Decision True := left I.
-Instance False_dec: Decision False := right (False_rect False).
-Instance Is_true_dec b : Decision (Is_true b).
-Proof. destruct b; simpl; apply _. Defined.
-
-Section prop_dec.
-  Context `(P_dec : Decision P) `(Q_dec : Decision Q).
-
-  Global Instance not_dec: Decision (¬P).
-  Proof. refine (cast_if_not P_dec); intuition. Defined.
-  Global Instance and_dec: Decision (P ∧ Q).
-  Proof. refine (cast_if_and P_dec Q_dec); intuition. Defined.
-  Global Instance or_dec: Decision (P ∨ Q).
-  Proof. refine (cast_if_or P_dec Q_dec); intuition. Defined.
-  Global Instance impl_dec: Decision (P → Q).
-  Proof. refine (if P_dec then cast_if Q_dec else left _); intuition. Defined.
-End prop_dec.
-Instance iff_dec `(P_dec : Decision P) `(Q_dec : Decision Q) :
-  Decision (P ↔ Q) := and_dec _ _.
-
-(** Instances of [Decision] for common data types. *)
-Instance bool_eq_dec : EqDecision bool.
-Proof. solve_decision. Defined.
-Instance unit_eq_dec : EqDecision unit.
-Proof. solve_decision. Defined.
-Instance Empty_set_eq_dec : EqDecision Empty_set.
-Proof. solve_decision. Defined.
-Instance prod_eq_dec `{EqDecision A, EqDecision B} : EqDecision (A * B).
-Proof. solve_decision. Defined.
-Instance sum_eq_dec `{EqDecision A, EqDecision B} : EqDecision (A + B).
-Proof. solve_decision. Defined.
-
-Instance curry_dec `(P_dec : ∀ (x : A) (y : B), Decision (P x y)) p :
-    Decision (curry P p) :=
-  match p as p return Decision (curry P p) with
-  | (x,y) => P_dec x y
-  end.
-
-Instance sig_eq_dec `(P : A → Prop) `{∀ x, ProofIrrel (P x), EqDecision A} :
-  EqDecision (sig P).
-Proof.
- refine (λ x y, cast_if (decide (`x = `y))); rewrite sig_eq_pi; trivial.
-Defined.
-
-(** Some laws for decidable propositions *)
-Lemma not_and_l {P Q : Prop} `{Decision P} : ¬(P ∧ Q) ↔ ¬P ∨ ¬Q.
-Proof. destruct (decide P); tauto. Qed.
-Lemma not_and_r {P Q : Prop} `{Decision Q} : ¬(P ∧ Q) ↔ ¬P ∨ ¬Q.
-Proof. destruct (decide Q); tauto. Qed.
-Lemma not_and_l_alt {P Q : Prop} `{Decision P} : ¬(P ∧ Q) ↔ ¬P ∨ (¬Q ∧ P).
-Proof. destruct (decide P); tauto. Qed.
-Lemma not_and_r_alt {P Q : Prop} `{Decision Q} : ¬(P ∧ Q) ↔ (¬P ∧ Q) ∨ ¬Q.
-Proof. destruct (decide Q); tauto. Qed.
-
-Program Definition inj_eq_dec `{EqDecision A} {B} (f : B → A)
-  `{!Inj (=) (=) f} : EqDecision B := λ x y, cast_if (decide (f x = f y)).
-Solve Obligations with firstorder congruence.
-
-(** * Instances of [RelDecision] *)
-Definition flip_dec {A} (R : relation A) `{!RelDecision R} :
-  RelDecision (flip R) := λ x y, decide_rel R y x.
-(** We do not declare this as an actual instance since Coq can unify [flip ?R]
-with any relation. Coq's standard library is carrying out the same approach for
-the [Reflexive], [Transitive], etc, instance of [flip]. *)
-Hint Extern 3 (RelDecision (flip _)) => apply flip_dec : typeclass_instances.

stdpp/dune

+(include_subdirs qualified)
+(coq.theory
+ (name stdpp)
+ (package coq-stdpp))

theories/fin.v → stdpp/fin.v

@@ -15,15 +15,15 @@ ambiguity. *)
 Notation fin := Fin.t.
 Notation FS := Fin.FS.
 
+Declare Scope fin_scope.
 Delimit Scope fin_scope with fin.
-Arguments Fin.FS _ _%fin : assert.
+Bind Scope fin_scope with fin.
+Global Arguments Fin.FS _ _%fin : assert.
 
-Notation "0" := Fin.F1 : fin_scope. Notation "1" := (FS 0) : fin_scope.
-Notation "2" := (FS 1) : fin_scope. Notation "3" := (FS 2) : fin_scope.
-Notation "4" := (FS 3) : fin_scope. Notation "5" := (FS 4) : fin_scope.
-Notation "6" := (FS 5) : fin_scope. Notation "7" := (FS 6) : fin_scope.
-Notation "8" := (FS 7) : fin_scope. Notation "9" := (FS 8) : fin_scope.
-Notation "10" := (FS 9) : fin_scope.
+(** Allow any non-negative number literal to be parsed as a [fin]. For example
+[42%fin : fin 64], or [42%fin : fin _], or [42%fin : fin (43 + _)]. *)
+Number Notation fin Nat.of_num_uint Nat.to_num_uint (via nat
+  mapping [[Fin.F1] => O, [Fin.FS] => S]) : fin_scope.
 
 Fixpoint fin_to_nat {n} (i : fin n) : nat :=
   match i with 0%fin => 0 | FS i => S (fin_to_nat i) end.
@@ -32,7 +32,7 @@ Coercion fin_to_nat : fin >-> nat.
 Notation nat_to_fin := Fin.of_nat_lt.
 Notation fin_rect2 := Fin.rect2.
 
-Instance fin_dec {n} : EqDecision (fin n).
+Global Instance fin_dec {n} : EqDecision (fin n).
 Proof.
  refine (fin_rect2
   (λ n (i j : fin n), { i = j } + { i ≠ j })
@@ -64,17 +64,19 @@ Ltac inv_fin i :=
   | fin ?n =>
     match eval hnf in n with
     | 0 =>
-      revert dependent i; match goal with |- ∀ i, @?P i => apply (fin_0_inv P) end
+      generalize dependent i;
+      match goal with |- ∀ i, @?P i => apply (fin_0_inv P) end
     | S ?n =>
-      revert dependent i; match goal with |- ∀ i, @?P i => apply (fin_S_inv P) end
+      generalize dependent i;
+      match goal with |- ∀ i, @?P i => apply (fin_S_inv P) end
     end
   end.
 
-Instance FS_inj: Inj (=) (=) (@FS n).
-Proof. intros n i j. apply Fin.FS_inj. Qed.
-Instance fin_to_nat_inj : Inj (=) (=) (@fin_to_nat n).
+Global Instance FS_inj {n} : Inj (=) (=) (@FS n).
+Proof. intros i j. apply Fin.FS_inj. Qed.
+Global Instance fin_to_nat_inj {n} : Inj (=) (=) (@fin_to_nat n).
 Proof.
-  intros n i. induction i; intros j; inv_fin j; intros; f_equal/=; auto with lia.
+  intros i. induction i; intros j; inv_fin j; intros; f_equal/=; auto with lia.
 Qed.
 
 Lemma fin_to_nat_lt {n} (i : fin n) : fin_to_nat i < n.
@@ -86,26 +88,26 @@ Qed.
 Lemma nat_to_fin_to_nat {n} (i : fin n) H : @nat_to_fin (fin_to_nat i) n H = i.
 Proof. apply (inj fin_to_nat), fin_to_nat_to_fin. Qed.
 
-Fixpoint fin_plus_inv {n1 n2} : ∀ (P : fin (n1 + n2) → Type)
+Fixpoint fin_add_inv {n1 n2} : ∀ (P : fin (n1 + n2) → Type)
     (H1 : ∀ i1 : fin n1, P (Fin.L n2 i1))
     (H2 : ∀ i2, P (Fin.R n1 i2)) (i : fin (n1 + n2)), P i :=
   match n1 with
   | 0 => λ P H1 H2 i, H2 i
-  | S n => λ P H1 H2, fin_S_inv P (H1 0%fin) (fin_plus_inv _ (λ i, H1 (FS i)) H2)
+  | S n => λ P H1 H2, fin_S_inv P (H1 0%fin) (fin_add_inv _ (λ i, H1 (FS i)) H2)
   end.
 
-Lemma fin_plus_inv_L {n1 n2} (P : fin (n1 + n2) → Type)
+Lemma fin_add_inv_l {n1 n2} (P : fin (n1 + n2) → Type)
     (H1: ∀ i1 : fin n1, P (Fin.L _ i1)) (H2: ∀ i2, P (Fin.R _ i2)) (i: fin n1) :
-  fin_plus_inv P H1 H2 (Fin.L n2 i) = H1 i.
+  fin_add_inv P H1 H2 (Fin.L n2 i) = H1 i.
 Proof.
   revert P H1 H2 i.
   induction n1 as [|n1 IH]; intros P H1 H2 i; inv_fin i; simpl; auto.
   intros i. apply (IH (λ i, P (FS i))).
 Qed.
 
-Lemma fin_plus_inv_R {n1 n2} (P : fin (n1 + n2) → Type)
+Lemma fin_add_inv_r {n1 n2} (P : fin (n1 + n2) → Type)
     (H1: ∀ i1 : fin n1, P (Fin.L _ i1)) (H2: ∀ i2, P (Fin.R _ i2)) (i: fin n2) :
-  fin_plus_inv P H1 H2 (Fin.R n1 i) = H2 i.
+  fin_add_inv P H1 H2 (Fin.R n1 i) = H2 i.
 Proof.
   revert P H1 H2 i; induction n1 as [|n1 IH]; intros P H1 H2 i; simpl; auto.
   apply (IH (λ i, P (FS i))).

theories/hashset.v → stdpp/hashset.v

@@ -10,8 +10,8 @@ Record hashset {A} (hash : A → Z) := Hashset {
   hashset_prf :
     map_Forall (λ n l, Forall (λ x, hash x = n) l ∧ NoDup l) hashset_car
 }.
-Arguments Hashset {_ _} _ _ : assert.
-Arguments hashset_car {_ _} _ : assert.
+Global Arguments Hashset {_ _} _ _ : assert.
+Global Arguments hashset_car {_ _} _ : assert.
 
 Section hashset.
 Context `{EqDecision A} (hash : A → Z).
@@ -39,7 +39,7 @@ Qed.
 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) _.
+    let l' := list_intersection l k in guard (l' ≠ []);; Some l') m1 m2) _.
 Next Obligation.
   intros _ _ m1 Hm1 m2 Hm2 n l'. rewrite lookup_intersection_with_Some.
   intros (?&?&?&?&?); simplify_option_eq.
@@ -49,7 +49,7 @@ Qed.
 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) _.
+    let l' := list_difference l k in guard (l' ≠ []);; Some l') m1 m2) _.
 Next Obligation.
   intros _ _ m1 Hm1 m2 Hm2 n l'. rewrite lookup_difference_with_Some.
   intros [[??]|(?&?&?&?&?)]; simplify_option_eq; auto.
@@ -105,7 +105,7 @@ Proof.
   - 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' [??] Hm];
-      csimpl; inversion_clear 1 as [|?? Hn]; [constructor|].
+      csimpl; inv 1 as [|?? Hn]; [constructor|].
     apply NoDup_app; split_and?; eauto.
     setoid_rewrite elem_of_list_bind; intros x ? ([n' l']&?&?); simpl in *.
     assert (hash x = n ∧ hash x = n') as [??]; subst.
@@ -116,7 +116,7 @@ Proof.
 Qed.
 End hashset.
 
-Typeclasses Opaque hashset_elem_of.
+Global Typeclasses Opaque hashset_elem_of.
 
 Section remove_duplicates.
 Context `{EqDecision A} (hash : A → Z).
@@ -126,7 +126,7 @@ Definition remove_dups_fast (l : list A) : list A :=
   | [] => []
   | [x] => [x]
   | _ =>
-     let n : Z := length l in
+     let n : Z := Z.of_nat (length l) in
      elements (foldr (λ x, ({[ x ]} ∪.)) ∅ l :
        hashset (λ x, hash x `mod` (2 * n))%Z)
   end.
@@ -134,7 +134,7 @@ Lemma elem_of_remove_dups_fast l x : x ∈ remove_dups_fast l ↔ x ∈ l.
 Proof.
   destruct l as [|x1 [|x2 l]]; try reflexivity.
   unfold remove_dups_fast; generalize (x1 :: x2 :: l); clear l; intros l.
-  generalize (λ x, hash x `mod` (2 * length l))%Z; intros f.
+  generalize (λ x, hash x `mod` (2 * Z.of_nat (length l)))%Z; intros f.
   rewrite elem_of_elements; split.
   - revert x. induction l as [|y l IH]; intros x; simpl.
     { by rewrite elem_of_empty. }
@@ -152,6 +152,6 @@ Definition listset_normalize (X : listset A) : listset A :=
   let (l) := X in Listset (remove_dups_fast l).
 Lemma listset_normalize_correct X : listset_normalize X ≡ X.
 Proof.
-  destruct X as [l]. apply elem_of_equiv; intro; apply elem_of_remove_dups_fast.
+  destruct X as [l]. apply set_equiv; intro; apply elem_of_remove_dups_fast.
 Qed.
 End remove_duplicates.