(** In this file, we show how we can use the solution of the recursive domain equations to build a higher-order separation logic *) Require Import ModuRes.PreoMet ModuRes.MetricRec ModuRes.CBUltInst. Require Import ModuRes.Finmap ModuRes.Constr. Require Import ModuRes.PCM ModuRes.UPred. (* This interface keeps some of the details of the solution opaque *) Module Type WORLD_PROP (Res : PCM_T). (* PreProp: The solution to the recursive equation. Equipped with a discrete order. *) Parameter PreProp : cmtyp. Instance PProp_preo : preoType PreProp := disc_preo PreProp. Instance PProp_pcm : pcmType PreProp := disc_pcm PreProp. Instance PProp_ext : extensible PreProp := disc_ext PreProp. (* Defines Worlds, Propositions *) Definition Wld := nat -f> PreProp. Definition Props := Wld -m> UPred Res.res. (* Define all the things on Props, so they have names - this shortens the terms later. *) Instance Props_ty : Setoid Props | 1 := _. Instance Props_m : metric Props | 1 := _. Instance Props_cm : cmetric Props | 1 := _. Instance Props_preo : preoType Props| 1 := _. Instance Props_pcm : pcmType Props | 1 := _. (* Establish the recursion isomorphism *) Parameter ı : PreProp -n> halve (cmfromType Props). Parameter ı' : halve (cmfromType Props) -n> PreProp. Axiom iso : forall P, ı' (ı P) == P. Axiom isoR: forall T, ı (ı' T) == T. End WORLD_PROP. (* Now we come to the actual implementation *) Module WorldProp (Res : PCM_T) : WORLD_PROP Res. (** The construction is parametric in the monoid we choose *) Import Res. (** We need to build a functor that would describe the following recursive domain equation: Prop ≃ (Loc -f> Prop) -m> UPred (Res) As usual, we split the negative and (not actually occurring) positive occurrences of Prop. *) Section Definitions. (** We'll be working with complete metric spaces, so whenever something needs an additional preorder, we'll just take a discrete one. *) Local Instance pt_disc P `{cmetric P} : preoType P | 2000 := disc_preo P. Local Instance pcm_disc P `{cmetric P} : pcmType P | 2000 := disc_pcm P. Definition FProp P `{cmP : cmetric P} := (nat -f> P) -m> UPred res. Context {U V} `{cmU : cmetric U} `{cmV : cmetric V}. Definition PropMorph (m : V -n> U) : FProp U -n> FProp V := fdMap (disc_m m) ▹. End Definitions. Module F <: SimplInput (CBUlt). Import CBUlt. Open Scope cat_scope. Definition F (T1 T2 : cmtyp) := cmfromType (FProp T1). Program Instance FArr : BiFMap F := fun P1 P2 P3 P4 => n[(PropMorph)] F 1 1 := umconst (pcmconst (up_cr (const True))). End F. Module F_In := InputHalve(F). Module Import Fix := Solution(CBUlt)(F_In). (** Now we can name the two isomorphic spaces of propositions, and the space of worlds. We'll store the actual solutions in the worlds, and use the action of the FPropO on them as the space we normally work with. *) Definition PreProp := DInfO. Definition Props := FProp PreProp. Definition Wld := (nat -f> PreProp). (* Define an order on PreProp. *) Instance PProp_preo: preoType PreProp := disc_preo PreProp. Instance PProp_pcm : pcmType PreProp := disc_pcm PreProp. Instance PProp_ext : extensible PreProp := disc_ext PreProp. (* Give names to the things for Props, so the terms can get shorter. *) Instance Props_ty : Setoid Props := _. Instance Props_m : metric Props := _. Instance Props_cm : cmetric Props := _. Instance Props_preo : preoType Props := _. Instance Props_pcm : pcmType Props := _. (* Establish the isomorphism *) Definition ı : PreProp -t> halve (cmfromType Props) := Unfold. Definition ı' : halve (cmfromType Props) -t> PreProp := Fold. Lemma iso P : ı' (ı P) == P. Proof. apply (FU_id P). Qed. Lemma isoR T : ı (ı' T) == T. Proof. apply (UF_id T). Qed. End WorldProp.