Move the lemmas from ds.v out to std++

_CoqProject

 -Q theories iris_logrel
 -arg -w -arg -notation-overridden,-redundant-canonical-projection,-several-object-files
-theories/prelude/ds.v
 theories/prelude/base.v
 theories/prelude/hax.v
 theories/F_mu_ref_conc/binder.v

theories/F_mu_ref_conc/binder.v

@@ -7,7 +7,7 @@ Inductive binder := BAnon | BNamed : string → binder.
 Delimit Scope binder_scope with bind.
 Bind Scope binder_scope with binder.
 
-Definition cons_binder (mx : binder) (X : gset string) : gset string :=
+Definition cons_binder (mx : binder) (X : stringset) : stringset :=
   match mx with BAnon => X | BNamed x => {[x]} ∪ X end.
 Infix ":b:" := cons_binder (at level 60, right associativity).
 Instance binder_eq_dec_eq : EqDecision binder.
@@ -45,7 +45,7 @@ Instance singleton_binder_set : Singleton binder stringset :=
 
 (** Properties lifts *)
 Lemma dom_delete_binder {A} (i : binder) (m : stringmap A) :
-  dom (gset string) (delete i m) ≡ (dom (gset string) m) ∖ {[i]}.
+  dom stringset (delete i m) ≡ (dom stringset m) ∖ {[i]}.
 Proof.
   destruct i; cbn-[difference]; eauto.
   - set_solver.
@@ -63,7 +63,7 @@ Lemma dom_insert_binder {A} (m : stringmap A) (i : binder) (x : A) :
   dom _ (<[i:=x]> m) = {[i]} ∪ dom _ m.
 Proof. destruct i; cbn-[union]; unfold_leibniz. set_solver.
        apply dom_insert. Qed.
-Lemma cons_binder_union (i : binder) (X : gset string) :
+Lemma cons_binder_union (i : binder) (X : stringset) :
   i :b: X = {[i]} ∪ X.
 Proof. destruct i; cbn-[union]; eauto. set_solver. Qed.
 

theories/F_mu_ref_conc/lang.v

@@ -66,7 +66,7 @@ Module lang.
   Proof. solve_decision. Defined.
 
   (** ** Closed expressions *)
-  Fixpoint is_closed (X : gset string) (e : expr) : bool :=
+  Fixpoint is_closed (X : stringset) (e : expr) : bool :=
     match e with
     | Var x => bool_decide (x ∈ X)
     | Rec f x e => is_closed (x :b: f :b: X) e
@@ -83,7 +83,7 @@ Module lang.
     | Unpack e e1 => is_closed X e && is_closed X e1
     end.
 
-  Class Closed (X : gset string) (e : expr) := closed : is_closed X e.
+  Class Closed (X : stringset) (e : expr) := closed : is_closed X e.
   Instance closed_proof_irrel env e : ProofIrrel (Closed env e).
   Proof. rewrite /Closed. apply _. Qed.
   Instance closed_decision env e : Decision (Closed env e).

theories/F_mu_ref_conc/reflection.v

@@ -119,7 +119,7 @@ Ltac of_expr e :=
     end
   end.
 
-Fixpoint is_closed (X : gset string) (e : expr) : bool :=
+Fixpoint is_closed (X : stringset) (e : expr) : bool :=
   match e with
   | Val _ _ _ => true
   | ClosedExpr e _ => true

theories/F_mu_ref_conc/subst.v

@@ -111,7 +111,7 @@ Proof.
 Qed.
 
 (** ** Properties of [Closed] and [is_closed] *)
-Lemma is_closed_weaken (e : expr) : ∀ (X Y : gset string),
+Lemma is_closed_weaken (e : expr) : ∀ (X Y : stringset),
     is_closed X e → X ⊆ Y → is_closed Y e.
 Proof. 
   induction e; intros X Y HXY HXe; simpl; simpl in HXe, HXY; eauto;
@@ -166,7 +166,7 @@ Proof.
   | [ |- Is_true (?P && ?Q) ] => apply andb_prop_intro; split; eauto
          end;
   try match goal with
-  | [ H : ∀ X : gset string,
+  | [ H : ∀ X : stringset,
         Closed X ?e → Closed X es → Closed X (subst x es ?e) 
         |- Is_true (is_closed _ (subst x es ?e)) ] =>
     eapply H; eauto; try (eapply Closed_mono; eauto; set_solver)

theories/F_mu_ref_conc/typing.v

@@ -166,7 +166,7 @@ Proof.
 Qed.
   
 Lemma typed_X_closed Γ τ e :
-  Γ ⊢ₜ e : τ → Closed (dom (gset string) Γ) e.
+  Γ ⊢ₜ e : τ → Closed (dom (stringset) Γ) e.
 Proof.
   intros H. rewrite /Closed. induction H => /=; simpl in *; auto with f_equal.
   - case_bool_decide; auto.

theories/logrel/fundamental_binary.v

@@ -140,16 +140,16 @@ Section masked.
       destruct x as [|x], f as [|f]; cbn-[union difference].
       + set_solver.
       + rewrite difference_empty.
-        rewrite assoc. rewrite difference_union_id.
+        rewrite assoc. rewrite difference_union.
         set_solver.
       + rewrite difference_empty.
-        rewrite assoc. rewrite difference_union_id.
+        rewrite assoc. rewrite difference_union.
         set_solver.
       + rewrite ?(right_id ∅ union).
         rewrite (comm union {[x]} {[f]}) !assoc.
-        rewrite difference_union_id.
+        rewrite difference_union.
         rewrite -!assoc (comm union {[f]} {[x]}) !assoc.
-        rewrite difference_union_id.
+        rewrite difference_union.
         set_solver. }
     assert (Closed (x :b: f :b: ∅)
               (subst_p (delete f (delete x (snd <$> vvs))) e')).
@@ -159,16 +159,16 @@ Section masked.
       destruct x as [|x], f as [|f]; cbn-[union difference].
       + set_solver.
       + rewrite difference_empty.
-        rewrite assoc. rewrite difference_union_id.
+        rewrite assoc. rewrite difference_union.
         set_solver.
       + rewrite difference_empty.
-        rewrite assoc. rewrite difference_union_id.
+        rewrite assoc. rewrite difference_union.
         set_solver.
       + rewrite ?(right_id ∅ union).
         rewrite (comm union {[x]} {[f]}) !assoc.
-        rewrite difference_union_id.
+        rewrite difference_union.
         rewrite -!assoc (comm union {[f]} {[x]}) !assoc.
-        rewrite difference_union_id.
+        rewrite difference_union.
         set_solver.
     }
     iModIntro. value_case'; eauto.

theories/prelude/base.v

@@ -2,7 +2,7 @@ From iris.algebra Require Export base.
 From iris.base_logic Require Import upred.
 From iris.program_logic Require Import weakestpre.
 From iris.base_logic Require Import invariants.
-From iris_logrel.prelude Require Export ds.
+From stdpp Require Export strings stringmap gmap mapset fin_maps.
 From Autosubst Require Export Autosubst.
 Import uPred.
 

theories/prelude/ds.v

-(* Data structures properties *)
-(* TODO: move this to std++   *)
-From stdpp Require Export strings gmap mapset stringmap.
-From iris.algebra Require Export base. (* for ssreflect stuff *)
-
-Lemma delete_insert_delete {A} (m : stringmap A) (i : string) (x : A) :
-  delete i (<[i:=x]> m) = delete i m.
-Proof.
-  apply map_eq=>j.
-  destruct (decide (i = j)) as [Eij|Nij];
-    simplify_map_eq; auto.
-Qed.
-
-Lemma delete_idem {A} (x : string) (m : stringmap A) :
-  delete x (delete x m) = delete x m.
-Proof.
-  rewrite delete_notin; first done.
-  apply lookup_delete.
-Qed.
-
-Lemma delete_singleton_ne {A} (i j : string) (x : A) :
-  j ≠ i →
-  delete i {[j := x]} = {[j := x]}.
-Proof. intros Hij. apply delete_notin. by apply lookup_singleton_ne. Qed.
-
-Lemma difference_empty_map {K A} `{EqDecision K} `{Countable K} (X : gmap K A) :
-  X ∖ ∅ = X.
-Proof.
-  apply map_eq => i.
-  remember (X !! i) as Z. destruct Z.
-  - apply lookup_difference_Some. split; eauto.
-  - apply lookup_difference_None. left; eauto.
-Qed.
-
-Lemma singleton_union_difference (X Y : stringset) (x : string) :
-  {[x]} ∪ (X ∖ Y) = ({[x]} ∪ X) ∖ (Y ∖ {[x]}).
-Proof.
-    unfold_leibniz. intros y. split; intro Hy. 
-    - apply elem_of_union in Hy. set_solver.
-    - apply elem_of_difference in Hy. destruct Hy as [Hy1 Hy2].
-      apply elem_of_union in Hy1.          
-      rewrite elem_of_union. rewrite elem_of_difference.
-      rewrite elem_of_singleton.
-      destruct (decide (x = y)); subst; eauto.
-      assert (y ∉ {[x]}). intro K; apply elem_of_singleton in K. auto.
-      right. destruct Hy1; set_solver.
-Qed.
-Lemma difference_union_id {A : Type} `{EqDecision A, Countable A} (X Y : gset A):
-  X ∖ Y ∪ Y = X ∪ Y.
-Proof.
-  unfold_leibniz. intro x.
-  rewrite !elem_of_union elem_of_difference.
-  split.
-  - set_solver.
-  - destruct (decide (x ∈ Y)); set_solver.
-Qed.
-Lemma difference_empty {A: Type} `{EqDecision A, Countable A} (X : gset A) :
-  X ∖ ∅ = X.
-Proof.
-  unfold_leibniz.
-  rewrite <- (right_id ∅ union (X ∖ ∅)).
-  rewrite <- (right_id ∅ union X) at 2.
-  fold_leibniz.
-  apply difference_union_id.
-Qed.
-