Library iris.base_logic.lib.iprop

From iris.algebra Require Import gmap.
From iris.algebra Require cofe_solver.
From iris.base_logic Require Export base_logic.
From iris.prelude Require Import options.

In this file we construct the type iProp of propositions of the Iris logic. This is done by solving the following recursive domain equation:
iProp ≈ uPred (∀ i : gid, gname -fin-> (Σ i) iProp)
Σ : gFunctors := lists of locally constractive functors i : gid := indexes addressing individual functors in Σ γ : gname := ghost variable names
The Iris logic is parametrized by a list of locally contractive functors Σ from the category of COFEs to the category of CMRAs. These functors are instantiated with iProp, the type of Iris propositions, which allows one to construct impredicate CMRAs, such as invariants and stored propositions using the agreement CMRA.

Locally contractive functors

The type gFunctor bundles a functor from the category of COFEs to the category of CMRAs with a proof that it is locally contractive.
Structure gFunctor := GFunctor {
  gFunctor_F :> rFunctor;
  gFunctor_map_contractive : rFunctorContractive gFunctor_F;
Global Arguments GFunctor _ {_}.
Global Existing Instance gFunctor_map_contractive.
Add Printing Constructor gFunctor.

The type gFunctors describes the parameters Σ of the Iris logic: lists of gFunctors.
Note that gFunctors is isomorphic to list gFunctor, but defined in an alternative way to avoid universe inconsistencies with respect to the universe monomorphic list type.
Defining gFunctors as a dependent record instead of a sigT avoids other universe inconsistencies.
Record gFunctors := GFunctors {
  gFunctors_len : nat;
  gFunctors_lookup : fin gFunctors_len gFunctor

Definition gid (Σ : gFunctors) := fin (gFunctors_len Σ).

Definition gname := positive.
Canonical Structure gnameO := leibnizO gname.

The resources functor iResF Σ A := i : gid, gname -fin (Σ i) A.
We define functions for the empty list of functors, the singleton list of functors, and the append operator on lists of functors. These are used to compose gFunctors out of smaller pieces.
Module gFunctors.
  Definition nil : gFunctors := GFunctors 0 (fin_0_inv _).

  Definition singleton (F : gFunctor) : gFunctors :=
    GFunctors 1 (fin_S_inv (λ _, gFunctor) F (fin_0_inv _)).

  Definition app (Σ1 Σ2 : gFunctors) : gFunctors :=
    GFunctors (gFunctors_len Σ1 + gFunctors_len Σ2)
              (fin_add_inv _ (gFunctors_lookup Σ1) (gFunctors_lookup Σ2)).
End gFunctors.

Coercion gFunctors.singleton : gFunctor >-> gFunctors.
Notation "#[ ]" := gFunctors.nil (format "#[ ]").
Notation "#[ Σ1 ; .. ; Σn ]" :=
  ( Σ1 .. ( Σn gFunctors.nil) ..).


In order to make proofs in the Iris logic modular, they are not done with respect to some concrete list of functors Σ, but are instead parametrized by an arbitrary list of functors Σ that contains at least certain functors. For example, the lock library is parameterized by a functor Σ that should have the functors corresponding to the heap and the exclusive monoid to manage to lock invariant.
The contraints to can be expressed using the type class subG Σ1 Σ2, which expresses that the functors Σ1 are contained in Σ2.
Avoid trigger happy type class search: this line ensures that type class search is only triggered if the arguments of subG do not contain evars. Since instance search for subG is restrained, instances should persistently have subG as their first parameter to avoid loops. For example, the instances subG_authΣ and auth_discrete otherwise create a cycle that pops up arbitrarily.
Global Hint Mode subG ! + : typeclass_instances.

Lemma subG_inv Σ1 Σ2 Σ : subG ( Σ1 Σ2) Σ subG Σ1 Σ × subG Σ2 Σ.
  moveH; split.
  - movei; move: H⇒ /(_ (Fin.L _ i)) [j] /=. rewrite fin_add_inv_l; eauto.
  - movei; move: H⇒ /(_ (Fin.R _ i)) [j] /=. rewrite fin_add_inv_r; eauto.

Global Instance subG_refl Σ : subG Σ Σ.
Proof. movei; by i. Qed.
Global Instance subG_app_l Σ Σ1 Σ2 : subG Σ Σ1 subG Σ ( Σ1 Σ2).
  moveH i; move: H⇒ /(_ i) [j ?].
   (Fin.L _ j). by rewrite /= fin_add_inv_l.
Global Instance subG_app_r Σ Σ1 Σ2 : subG Σ Σ2 subG Σ ( Σ1 Σ2).
  moveH i; move: H⇒ /(_ i) [j ?].
   (Fin.R _ j). by rewrite /= fin_add_inv_r.

Solution of the recursive domain equation

We first declare a module type and then an instance of it so as to seal all of the construction, this way we are sure we do not use any properties of the construction, and also avoid Coq from blindly unfolding it.
Module Type iProp_solution_sig.
  Parameter iPrePropO : gFunctors ofe.
  Global Declare Instance iPreProp_cofe {Σ} : Cofe (iPrePropO Σ).

  Definition iResUR (Σ : gFunctors) : ucmra :=
    discrete_funUR (λ i,
      gmapUR gname (rFunctor_apply (gFunctors_lookup Σ i) (iPrePropO Σ))).
  Notation iProp Σ := (uPred (iResUR Σ)).
  Notation iPropO Σ := (uPredO (iResUR Σ)).
  Notation iPropI Σ := (uPredI (iResUR Σ)).

  Parameter iProp_unfold: {Σ}, iPropO Σ -n> iPrePropO Σ.
  Parameter iProp_fold: {Σ}, iPrePropO Σ -n> iPropO Σ.
  Parameter iProp_fold_unfold: {Σ} (P : iProp Σ),
    iProp_fold (iProp_unfold P) P.
  Parameter iProp_unfold_fold: {Σ} (P : iPrePropO Σ),
    iProp_unfold (iProp_fold P) P.
End iProp_solution_sig.

Module Export iProp_solution : iProp_solution_sig.
  Import cofe_solver.
  Definition iProp_result (Σ : gFunctors) :
    solution (uPredOF (iResF Σ)) := solver.result _.
  Definition iPrePropO (Σ : gFunctors) : ofe := iProp_result Σ.
  Global Instance iPreProp_cofe {Σ} : Cofe (iPrePropO Σ) := _.

  Definition iResUR (Σ : gFunctors) : ucmra :=
    discrete_funUR (λ i,
      gmapUR gname (rFunctor_apply (gFunctors_lookup Σ i) (iPrePropO Σ))).
  Notation iProp Σ := (uPred (iResUR Σ)).
  Notation iPropO Σ := (uPredO (iResUR Σ)).

  Definition iProp_unfold {Σ} : iPropO Σ -n> iPrePropO Σ :=
    ofe_iso_1 (iProp_result Σ).
  Definition iProp_fold {Σ} : iPrePropO Σ -n> iPropO Σ :=
    ofe_iso_2 (iProp_result Σ).
  Lemma iProp_fold_unfold {Σ} (P : iProp Σ) : iProp_fold (iProp_unfold P) P.
  Proof. apply ofe_iso_21. Qed.
  Lemma iProp_unfold_fold {Σ} (P : iPrePropO Σ) : iProp_unfold (iProp_fold P) P.
  Proof. apply ofe_iso_12. Qed.
End iProp_solution.

Properties of the solution to the recursive domain equation

Lemma iProp_unfold_equivI {Σ} (P Q : iProp Σ) :
  iProp_unfold P iProp_unfold Q ⊢@{iPropI Σ} P Q.
  rewrite -{2}(iProp_fold_unfold P) -{2}(iProp_fold_unfold Q). apply: f_equivI.