Move global functor construction to its own file and define notations.

And now the part that I forgot to commit.

_CoqProject

@@ -67,6 +67,7 @@ program_logic/hoare.v
 program_logic/language.v
 program_logic/tests.v
 program_logic/ghost_ownership.v
+program_logic/global_functor.v
 program_logic/saved_prop.v
 program_logic/auth.v
 program_logic/sts.v

barrier/barrier.v

@@ -125,7 +125,7 @@ End barrier_proto.
 Import barrier_proto.
 
 (* The functors we need. *)
-Definition barrierGFs := stsGF sts `::` agreeF `::` pnil.
+Definition barrierGFs : iFunctors := [stsGF sts; agreeF].
 
 (** Now we come to the Iris part of the proof. *)
 Section proof.

barrier/client.v

@@ -26,7 +26,7 @@ Section client.
 End client.
 
 Section ClosedProofs.
-  Definition Σ : iFunctorG := heapGF .:: barrierGFs .++ endGF.
+  Definition Σ : iFunctorG := #[ heapGF ; barrierGFs ].
   Notation iProp := (iPropG heap_lang Σ).
 
   Lemma client_safe_closed σ : {{ ownP σ : iProp }} client {{ λ v, True }}.

heap_lang/tests.v

@@ -76,7 +76,7 @@ Section LiftingTests.
 End LiftingTests.
 
 Section ClosedProofs.
-  Definition Σ : iFunctorG := heapGF .:: endGF.
+  Definition Σ : iFunctorG := #[ heapGF ].
   Notation iProp := (iPropG heap_lang Σ).
 
   Lemma heap_e_closed σ : {{ ownP σ : iProp }} heap_e {{ λ v, v = '2 }}.

prelude/functions.v

@@ -27,45 +27,4 @@ Section functions.
   Lemma fn_lookup_alter_ne (g : T → T) (f : A → T) a b :
     a ≠ b → alter g a f b = f b.
   Proof. unfold alter, fn_alter. by destruct (decide (a = b)). Qed.
-
 End functions.
-
-(** "Cons-ing" of functions from nat to T *)
-(* Coq's standard lists are not universe polymorphic. Hence we have to re-define them. Ouch.
-   TODO: If we decide to end up going with this, we should move this elsewhere. *)
-Polymorphic Inductive plist {A : Type} : Type :=
-| pnil : plist
-| pcons: A → plist → plist.
-Arguments plist : clear implicits.
-
-Polymorphic Fixpoint papp {A : Type} (l1 l2 : plist A) : plist A :=
-  match l1 with
-  | pnil => l2
-  | pcons a l => pcons a (papp l l2)
-  end.
-
-(* TODO: Notation is totally up for debate. *)
-Infix "`::`" := pcons (at level 60, right associativity) : C_scope.
-Infix "`++`" := papp (at level 60, right associativity) : C_scope.
-
-Polymorphic Definition fn_cons {T : Type} (t : T) (f: nat → T) : nat → T :=
-  λ n, match n with
-       | O => t
-       | S n => f n
-       end.
-
-Polymorphic Fixpoint fn_mcons {T : Type} (ts : plist T) (f : nat → T) : nat → T :=
-  match ts with
-  | pnil => f
-  | pcons t ts => fn_cons t (fn_mcons ts f)
-  end.
-
-(* TODO: Notation is totally up for debate. *)
-Infix ".::" := fn_cons (at level 60, right associativity) : C_scope.
-Infix ".++" := fn_mcons (at level 60, right associativity) : C_scope.
-
-Polymorphic Lemma fn_mcons_app {T : Type} (ts1 ts2 : plist T) f :
-  (ts1 `++` ts2) .++ f = ts1 .++ (ts2 .++ f).
-Proof.
-  induction ts1; simpl; eauto. congruence.
-Qed.

program_logic/auth.v

 From algebra Require Export auth.
-From program_logic Require Export invariants ghost_ownership.
+From program_logic Require Export invariants global_functor.
 Import uPred.
 
 Class authG Λ Σ (A : cmraT) `{Empty A} := AuthG {

program_logic/ghost_ownership.v

@@ -6,13 +6,17 @@ Import uPred.
 
 (** Index of a CMRA in the product of global CMRAs. *)
 Definition gid := nat.
+
 (** Name of one instance of a particular CMRA in the ghost state. *)
 Definition gname := positive.
+
 (** The global CMRA: Indexed product over a gid i to (gname --fin--> Σ i) *)
 Definition globalF (Σ : gid → iFunctor) : iFunctor :=
   iprodF (λ i, mapF gname (Σ i)).
+Notation iFunctorG := (gid → iFunctor).
+Notation iPropG Λ Σ := (iProp Λ (globalF Σ)).
 
-Class inG (Λ : language) (Σ : gid → iFunctor) (A : cmraT) := InG {
+Class inG (Λ : language) (Σ : iFunctorG) (A : cmraT) := InG {
   inG_id : gid;
   inG_prf : A = Σ inG_id (laterC (iPreProp Λ (globalF Σ)))
 }.
@@ -25,32 +29,6 @@ Instance: Params (@to_globalF) 5.
 Instance: Params (@own) 5.
 Typeclasses Opaque to_globalF own.
 
-Notation iPropG Λ Σ := (iProp Λ (globalF Σ)).
-Notation iFunctorG := (gid → iFunctor).
-
-(** We need another typeclass to identify the *functor* in the Σ. Basing inG on
-   the functor breaks badly because Coq is unable to infer the correct
-   typeclasses, it does not unfold the functor. *)
-Class inGF (Λ : language) (Σ : gid → iFunctor) (F : iFunctor) := InGF {
-  inGF_id : gid;
-  inGF_prf : F = Σ inGF_id;
-}.
-(* Avoid eager type class search: this line ensures that type class search
-is only triggered if the first two arguments of inGF do not contain evars. Since
-instance search for [inGF] is restrained, instances should always have [inGF] as
-their first argument to avoid loops. For example, the instances [authGF_inGF]
-and [auth_identity] otherwise create a cycle that pops up arbitrarily. *)
-Hint Mode inGF + + - : typeclass_instances.
-
-Lemma inGF_inG `{inGF Λ Σ F} : inG Λ Σ (F (laterC (iPreProp Λ (globalF Σ)))).
-Proof. exists inGF_id. by rewrite -inGF_prf. Qed.
-Instance inGF_here {Λ Σ} (F: iFunctor) : inGF Λ (F .:: Σ) F.
-Proof. by exists 0. Qed.
-Instance inGF_further {Λ Σ} (F F': iFunctor) : inGF Λ Σ F → inGF Λ (F' .:: Σ) F.
-Proof. intros [i ?]. by exists (S i). Qed.
-
-Definition endGF : iFunctorG := const (constF unitRA).
-
 (** Properties about ghost ownership *)
 Section global.
 Context `{i : inG Λ Σ A}.

program_logic/saved_prop.v

 From algebra Require Export agree.
-From program_logic Require Export ghost_ownership.
+From program_logic Require Export global_functor.
 Import uPred.
 
 Notation savedPropG Λ Σ :=

program_logic/sts.v

 From algebra Require Export sts.
-From program_logic Require Export invariants ghost_ownership.
+From program_logic Require Export invariants global_functor.
 Import uPred.
 
 Class stsG Λ Σ (sts : stsT) := StsG {