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lifted_modalities.v
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lifted_modalities.v
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(** Here we "lift" the various modalities in Perennial from [iProp] to
[dProp]. This is mostly just boilerplate. *)
From iris.bi Require Import derived_laws.
From iris.proofmode Require Import base tactics classes.
From Perennial.base_logic.lib Require Import ncfupd.
From Perennial.program_logic Require Import crash_weakestpre cfupd.
From self.algebra Require Import ghost_map.
From self Require Import extra.
From self.base Require Import primitive_laws class_instances.
From self.high Require Export dprop resources monpred_simpl
post_crash_modality increasing_map state_interpretation wpc_notation.
From self.base Require Import primitive_laws class_instances.
From self.high Require Import dprop monpred_simpl.
Program Definition uPred_fupd_split_level_def `{!invGS Σ}
(E1 E2 : coPset) (k : nat) mj (P : dProp Σ) : dProp Σ :=
MonPred (λ TV, uPred_fupd_split_level_def E1 E2 k mj (P TV))%I _.
Next Obligation. solve_proper. Qed.
Definition uPred_fupd_split_level_aux `{!invGS Σ} : seal uPred_fupd_split_level_def.
Proof. by eexists. Qed.
Definition uPred_fupd_split_level `{!invGS Σ} := uPred_fupd_split_level_aux.(unseal).
Definition uPred_fupd_split_level_eq `{!invGS Σ} :
uPred_fupd_split_level = uPred_fupd_split_level_def :=
uPred_fupd_split_level_aux.(seal_eq).
Definition uPred_fupd_level_def `{!invGS Σ} (E1 E2 : coPset) (k : nat) (P : dProp Σ) : dProp Σ :=
uPred_fupd_split_level E1 E2 k None P.
Definition uPred_fupd_level_aux `{!invGS Σ} : seal uPred_fupd_level_def.
Proof. by eexists. Qed.
Definition uPred_fupd_level `{!invGS Σ} := uPred_fupd_level_aux.(unseal).
Definition uPred_fupd_level_eq `{!invGS Σ} : uPred_fupd_level = uPred_fupd_level_def :=
uPred_fupd_level_aux.(seal_eq).
Notation "| k , j ={ E1 , E2 }=> Q" := (uPred_fupd_split_level E1 E2 k j Q) : bi_scope.
Notation "| k , j ={ E1 }=> Q" := (uPred_fupd_split_level E1 E1 k j Q) : bi_scope.
Notation "| k ={ E1 , E2 }=> Q" := (uPred_fupd_level E1 E2 k Q) : bi_scope.
Notation "| k ={ E1 }=> Q" := (uPred_fupd_level E1 E1 k Q) : bi_scope.
Section lifted_fupd_level.
Context `{!invGS Σ}.
(*** fupd_level*)
Global Instance except_0_fupd_level' E1 E2 k P :
IsExcept0 (|k={E1,E2}=> P).
Proof. Abort. (* by rewrite /IsExcept0 except_0_fupd_level. Qed. *)
Global Instance from_modal_fupd_level E k P :
FromModal True modality_id (|k={E}=> P) (|k={E}=> P) P.
Proof. Abort. (* by rewrite /FromModal True /= -fupd_level_intro. Qed. *)
Global Instance elim_modal_bupd_fupd_level p E1 E2 k P Q :
ElimModal True p false (|==> P) P (|k={E1,E2}=> Q) (|k={E1,E2}=> Q) | 10.
Proof. Abort.
(* by rewrite /ElimModal intuitionistically_if_elim *)
(* (bupd_fupd_level E1 k) fupd_level_frame_r wand_elim_r fupd_level_trans. *)
(* Qed. *)
Global Instance elim_modal_fupd_level_fupd_level p E1 E2 E3 k P Q :
ElimModal True p false (|k={E1,E2}=> P) P (|k={E1,E3}=> Q) (|k={E2,E3}=> Q).
Proof. Abort.
(* by rewrite /ElimModal intuitionistically_if_elim *)
(* fupd_level_frame_r wand_elim_r fupd_level_trans. *)
(* Qed. *)
Lemma fupd_split_level_unfold_at E1 E2 k mj P TV :
(uPred_fupd_split_level E1 E2 k mj P) TV = fupd_level.uPred_fupd_split_level E1 E2 k mj (P TV).
Proof.
rewrite uPred_fupd_split_level_eq.
rewrite /uPred_fupd_split_level_def.
rewrite fupd_level.uPred_fupd_split_level_eq.
reflexivity.
Qed.
(* NOTE: This lemma may not be needed anymore. *)
Lemma fupd_level_unfold_at E1 E2 k P TV :
(uPred_fupd_level E1 E2 k P) TV = fupd_level.uPred_fupd_level E1 E2 k (P TV).
Proof.
rewrite uPred_fupd_level_eq.
rewrite /uPred_fupd_level_def.
rewrite fupd_level.uPred_fupd_level_eq.
apply fupd_split_level_unfold_at.
Qed.
End lifted_fupd_level.
Program Definition ncfupd_def `{!nvmG Σ} (E1 E2 : coPset) (P : dProp Σ) : dProp Σ :=
MonPred (λ i, let nD := i.2 in ncfupd E1 E2 (P i))%I _.
Next Obligation.
intros.
intros [??] [??] [? [= <-]].
simpl.
apply ncfupd_mono.
apply monPred_mono.
done.
Qed.
Definition ncfupd_aux `{!nvmG Σ} : seal (ncfupd_def). Proof. by eexists. Qed.
Definition ncfupd `{!nvmG Σ} := ncfupd_aux.(unseal).
Definition ncfupd_eq `{!nvmG Σ} : ncfupd = ncfupd_def := ncfupd_aux.(seal_eq).
Notation "|NC={ E1 }=> Q" := (ncfupd E1 E1 Q)
(at level 99, E1 at level 50, Q at level 200,
format "'[ ' |NC={ E1 }=> '/' Q ']'") : bi_scope.
Notation "|NC={ E1 , E2 }=> P" := (ncfupd E1 E2 P)
(at level 99, E1, E2 at level 50, P at level 200,
format "'[ ' |NC={ E1 , E2 }=> '/' P ']'") : bi_scope.
Notation "|NC={ Eo } [ Ei ]▷=> Q" := (∀ q, NC q -∗ |={Eo,Ei}=> ▷ |={Ei,Eo}=> Q ∗ NC q)%I
(at level 99, Eo, Ei at level 50, Q at level 200,
format "'[ ' |NC={ Eo } [ Ei ]▷=> '/' Q ']'") : bi_scope.
Notation "|NC={ E1 } [ E2 ]▷=>^ n Q" := (Nat.iter n (λ P, |NC={E1}[E2]▷=> P) Q)%I
(at level 99, E1, E2 at level 50, n at level 9, Q at level 200,
format "'[ ' |NC={ E1 } [ E2 ]▷=>^ n '/' Q ']'").
Program Definition cfupd `{!nvmG Σ} E1 (P : dProp Σ) :=
(with_gnames (λ nD, ⎡ C ⎤) -∗ |={E1}=> P)%I.
(* MonPred (λ TV, cfupd k E1 (P TV))%I _. *)
(* Next Obligation. solve_proper. Qed. *)
Notation "|C={ E1 }=> P" := (cfupd E1 P)
(at level 99, E1 at level 50, P at level 200,
format "'[ ' |C={ E1 }=> '/' P ']'").
Section lifted_modalities.
Context `{nvmG Σ}.
(*** ncfupd *)
Global Instance from_modal_ncfupd E P :
FromModal True modality_id (|NC={E}=> P) (|NC={E}=> P) P.
Proof.
rewrite /FromModal ncfupd_eq /=. iStartProof (iProp _). iIntros (_ TV).
iIntros "$". rewrite ncfupd.ncfupd_eq. iIntros (q) "$". done.
Qed.
Lemma ncfupd_unfold_at E1 E2 P TV gnames :
(ncfupd E1 E2 P) (TV, gnames) = ncfupd.ncfupd E1 E2 (P (TV, gnames)).
Proof. rewrite ncfupd_eq /ncfupd_def. reflexivity. Qed.
Global Instance elim_modal_bupd_ncfupd p E1 E2 P Q :
ElimModal True p false (|==> P) P (|NC={E1,E2}=> Q) (|NC={E1,E2}=> Q) | 10.
Proof.
rewrite /ElimModal.
intros ?.
iModel.
rewrite ncfupd_eq.
monPred_simpl. simpl.
rewrite bi.intuitionistically_if_elim (bupd_ncfupd E1).
rewrite ncfupd_frame_r.
iIntros "H".
iDestruct (ncfupd.ncfupd_mono with "H") as "H".
{ iIntros "[P i]". iSpecialize ("i" with "[] P"); first done. iApply "i". }
iApply ncfupd_trans.
done.
Qed.
Lemma ncfupd_frame_r E1 E2 P R:
(|NC={E1,E2}=> P) ∗ R ⊢ |NC={E1,E2}=> P ∗ R.
Proof.
iModel.
rewrite 2!ncfupd_unfold_at.
rewrite monPred_at_sep.
iApply ncfupd_frame_r.
Qed.
Lemma ncfupd_trans E1 E2 E3 P : (|NC={E1,E2}=> |NC={E2,E3}=> P) ⊢ |NC={E1,E3}=> P.
Proof.
iModel.
rewrite 3!ncfupd_unfold_at.
iApply ncfupd_trans.
Qed.
Global Instance ncfupd_ne E1 E2 :
NonExpansive (ncfupd E1 E2).
Proof. rewrite ncfupd_eq. split. intros TV. solve_proper. Qed.
Global Instance ncfupd_proper E1 E2 :
Proper ((≡) ==> (≡)) (ncfupd E1 E2) := ne_proper _.
Lemma ncfupd_mono E1 E2 P Q : (P ⊢ Q) → (|NC={E1,E2}=> P) ⊢ |NC={E1,E2}=> Q.
Proof.
intro wand.
iModel.
rewrite 2!ncfupd_unfold_at.
iApply ncfupd_mono.
iApply wand.
Qed.
Global Instance ncfupd_mono' E1 E2 : Proper ((⊢) ==> (⊢)) (ncfupd E1 E2).
Proof. intros P Q; apply ncfupd_mono. Qed.
Global Instance ncfupd_flip_mono' E1 E2 :
Proper (flip (⊢) ==> flip (⊢)) (ncfupd E1 E2).
Proof. intros P Q; apply ncfupd_mono. Qed.
Global Instance elim_modal_ncfupd_ncfupd p E1 E2 E3 P Q :
ElimModal True p false (|NC={E1,E2}=> P) P (|NC={E1,E3}=> Q) (|NC={E2,E3}=> Q).
Proof.
rewrite /ElimModal. rewrite bi.intuitionistically_if_elim.
rewrite ncfupd_frame_r. rewrite bi.wand_elim_r. rewrite ncfupd_trans.
done.
Qed.
Lemma fupd_ncfupd E1 E2 (P : dProp Σ) : (|={E1,E2}=> P) ⊢ |NC={E1,E2}=> P.
Proof.
iModel.
rewrite ncfupd_unfold_at.
iApply fupd_ncfupd.
Qed.
Global Instance elim_modal_fupd_ncfupd p E1 E2 E3 P Q :
ElimModal True p false (|={E1,E2}=> P) P (|NC={E1,E3}=> Q) (|NC={E2,E3}=> Q).
Proof.
rewrite /ElimModal => ?. rewrite (fupd_ncfupd _ _) bi.intuitionistically_if_elim
ncfupd_frame_r bi.wand_elim_r ncfupd_trans //=.
Qed.
(*** cfupd *)
Lemma cfupd_unfold_at E1 P TV gnames :
(cfupd E1 P) (TV, gnames) ⊣⊢ cfupd.cfupd E1 (P (TV, gnames)).
Proof.
rewrite /cfupd. rewrite /cfupd.cfupd.
monPred_simpl.
setoid_rewrite monPred_at_fupd.
setoid_rewrite monPred_at_embed.
iSplit.
- iIntros "H".
iSpecialize ("H" $! (TV, gnames)).
iApply "H".
done.
- iIntros "H". iIntros ([??] incl) "C".
iApply monPred_mono; first apply incl.
destruct incl as [? [= <-]].
iApply "H". iFrame.
Qed.
Global Instance from_modal_cfupd E1 P :
FromModal True modality_id (cfupd E1 P) (cfupd E1 P) (P).
Proof.
rewrite /FromModal /=.
intros _.
iModel.
rewrite cfupd_unfold_at.
iIntros "HP".
iModIntro.
iFrame.
Qed.
End lifted_modalities.
(** A few hints. These are declared outside of the section as Coq does not allow
adding global hints inside a section. *)
(** This hint makes [auto] work when the goal is behind a ncfupd modality. *)
Global Hint Extern 1 (environments.envs_entails _ (|NC={_}=> _)) => iModIntro : core.