From iris.base_logic Require Import base_logic.
From iris.proofmode Require Import tactics.
Set Default Proof Using "Type".
Import uPred.
(** The "core" of an assertion is its maximal persistent part.
It can be defined entirely within the logic... at least
in the shallow embedding.
WARNING: The function "coreP" is NOT NON-EXPANSIVE.
This is because the turnstile is not non-expansive as a function
from iProp to (discreteC Prop).
To obtain a core that's non-expansive, we would have to add another
modality to the logic: a box that removes access to *all* resources,
not just restricts access to the core.
*)
Definition coreP {M : ucmraT} (P : uPred M) : uPred M :=
(∀ `(!PersistentP Q), ⌜P ⊢ Q⌝ → Q)%I.
Instance: Params (@coreP) 1.
Typeclasses Opaque coreP.
Section core.
Context {M : ucmraT}.
Implicit Types P Q : uPred M.
Lemma coreP_intro P : P -∗ coreP P.
Proof. rewrite /coreP. iIntros "HP". by iIntros (Q HQ ->). Qed.
Global Instance coreP_persistent P : PersistentP (coreP P).
Proof. rewrite /coreP. apply _. Qed.
Global Instance coreP_mono : Proper ((⊢) ==> (⊢)) (@coreP M).
Proof.
rewrite /coreP. iIntros (P P' ?) "H"; iIntros (Q ??).
iApply ("H" $! Q with "[%]"). by etrans.
Qed.
Global Instance coreP_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@coreP M).
Proof. intros P Q. rewrite !equiv_spec=>-[??]. by split; apply coreP_mono. Qed.
Lemma coreP_elim P : PersistentP P → coreP P -∗ P.
Proof. rewrite /coreP. iIntros (?) "HCP". unshelve iApply ("HCP" $! P); auto. Qed.
Lemma coreP_wand P Q :
(coreP P ⊢ Q) ↔ (P ⊢ □ Q).
Proof.
split.
- iIntros (HP) "HP". iDestruct (coreP_intro with "HP") as "#HcP".
iAlways. by iApply HP.
- iIntros (HPQ) "HcP". iDestruct (coreP_mono _ _ HPQ with "HcP") as "HcQ".
iDestruct (coreP_elim with "HcQ") as "#HQ". done.
Qed.
End core.