world_prop.v 3.53 KB
 Filip Sieczkowski committed May 28, 2014 1 2 ``````(** In this file, we show how we can use the solution of the recursive domain equations to build a higher-order separation logic *) `````` Filip Sieczkowski committed Oct 07, 2014 3 4 5 ``````Require Import ModuRes.PreoMet ModuRes.MetricRec ModuRes.CBUltInst. Require Import ModuRes.Finmap ModuRes.Constr. Require Import ModuRes.PCM ModuRes.UPred ModuRes.BI. `````` Filip Sieczkowski committed May 28, 2014 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 `````` Module WorldProp (Res : PCM_T). (** 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} := `````` Filip Sieczkowski committed Jun 03, 2014 26 `````` (nat -f> P) -m> UPred res. `````` Filip Sieczkowski committed May 28, 2014 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 `````` 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). 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. `````` Ralf Jung committed Jan 31, 2015 83 84 85 86 87 `````` Set Printing All. (* PreProp has an equivalence and a complete metric. It also has a preorder that fits to everything else. *) Instance PProp_ty : Setoid PreProp := _. Instance PProp_m : metric PreProp := _. Instance PProp_cm : cmetric PreProp := _. `````` Filip Sieczkowski committed May 28, 2014 88 89 90 91 `````` Instance PProp_preo : preoType PreProp := disc_preo PreProp. Instance PProp_pcm : pcmType PreProp := disc_pcm PreProp. Instance PProp_ext : extensible PreProp := disc_ext PreProp. `````` Ralf Jung committed Jan 31, 2015 92 93 94 95 96 97 98 `````` (* 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 := _. `````` Filip Sieczkowski committed May 28, 2014 99 ``End WorldProp.``