Library iris.base_logic.lib.fancy_updates_from_vs
From stdpp Require Export coPset.
From iris.proofmode Require Import proofmode.
From iris.base_logic Require Export base_logic.
Set Default Proof Using "Type*".
Section fupd.
Context {M} (vs : coPset → coPset → uPred M → uPred M → uPred M).
Notation "P ={ E1 , E2 }=> Q" := (vs E1 E2 P Q)
(at level 99, E1,E2 at level 50, Q at level 200,
format "P ={ E1 , E2 }=> Q") : bi_scope.
Context (vs_ne : ∀ E1 E2, NonExpansive2 (vs E1 E2)).
Context (vs_persistent : ∀ E1 E2 P Q, Persistent (P ={E1,E2}=> Q)).
Context (vs_impl : ∀ E P Q, □ (P → Q) ⊢ P ={E,E}=> Q).
Context (vs_trans : ∀ E1 E2 E3 P Q R,
(P ={E1,E2}=> Q) ∧ (Q ={E2,E3}=> R) ⊢ P ={E1,E3}=> R).
Context (vs_mask_frame_r : ∀ E1 E2 Ef P Q,
E1 ## Ef → (P ={E1,E2}=> Q) ⊢ P ={E1 ∪ Ef,E2 ∪ Ef}=> Q).
Context (vs_frame_r : ∀ E1 E2 P Q R, (P ={E1,E2}=> Q) ⊢ P ∗ R ={E1,E2}=> Q ∗ R).
Context (vs_exists : ∀ {A} E1 E2 (Φ : A → uPred M) Q,
(∀ x, Φ x ={E1,E2}=> Q) ⊢ (∃ x, Φ x) ={E1,E2}=> Q).
Context (vs_persistent_intro_r : ∀ E1 E2 P Q R,
Persistent R →
(R -∗ (P ={E1,E2}=> Q)) ⊢ P ∗ R ={E1,E2}=> Q).
Definition fupd (E1 E2 : coPset) (P : uPred M) : uPred M :=
∃ R, R ∗ vs E1 E2 R P.
Notation "|={ E1 , E2 }=> Q" := (fupd E1 E2 Q) : bi_scope.
Global Instance fupd_ne E1 E2 : NonExpansive (@fupd E1 E2).
Proof. solve_proper. Qed.
Lemma fupd_intro E P : P ⊢ |={E,E}=> P.
Proof. iIntros "HP". iExists P. iFrame "HP". iApply vs_impl; auto. Qed.
Lemma fupd_mono E1 E2 P Q : (P ⊢ Q) → (|={E1,E2}=> P) ⊢ |={E1,E2}=> Q.
Proof.
iIntros (HPQ); iDestruct 1 as (R) "[HR Hvs]".
iExists R; iFrame "HR". iApply (vs_trans with "[$Hvs]").
iApply vs_impl. iIntros "!> HP". by iApply HPQ.
Qed.
Lemma fupd_trans E1 E2 E3 P : (|={E1,E2}=> |={E2,E3}=> P) ⊢ |={E1,E3}=> P.
Proof.
iDestruct 1 as (R) "[HR Hvs]". iExists R. iFrame "HR".
iApply (vs_trans with "[$Hvs]"). clear R.
iApply vs_exists; iIntros (R). iApply vs_persistent_intro_r; iIntros "Hvs".
iApply (vs_trans with "[$Hvs]"). iApply vs_impl; auto.
Qed.
Lemma fupd_mask_frame_r E1 E2 Ef P :
E1 ## Ef → (|={E1,E2}=> P) ⊢ |={E1 ∪ Ef,E2 ∪ Ef}=> P.
Proof.
iIntros (HE); iDestruct 1 as (R) "[HR Hvs]". iExists R; iFrame "HR".
by iApply vs_mask_frame_r.
Qed.
Lemma fupd_frame_r E1 E2 P Q : (|={E1,E2}=> P) ∗ Q ⊢ |={E1,E2}=> P ∗ Q.
Proof.
iIntros "[Hvs HQ]". iDestruct "Hvs" as (R) "[HR Hvs]".
iExists (R ∗ Q)%I. iFrame "HR HQ". by iApply vs_frame_r.
Qed.
End fupd.
From iris.proofmode Require Import proofmode.
From iris.base_logic Require Export base_logic.
Set Default Proof Using "Type*".
Section fupd.
Context {M} (vs : coPset → coPset → uPred M → uPred M → uPred M).
Notation "P ={ E1 , E2 }=> Q" := (vs E1 E2 P Q)
(at level 99, E1,E2 at level 50, Q at level 200,
format "P ={ E1 , E2 }=> Q") : bi_scope.
Context (vs_ne : ∀ E1 E2, NonExpansive2 (vs E1 E2)).
Context (vs_persistent : ∀ E1 E2 P Q, Persistent (P ={E1,E2}=> Q)).
Context (vs_impl : ∀ E P Q, □ (P → Q) ⊢ P ={E,E}=> Q).
Context (vs_trans : ∀ E1 E2 E3 P Q R,
(P ={E1,E2}=> Q) ∧ (Q ={E2,E3}=> R) ⊢ P ={E1,E3}=> R).
Context (vs_mask_frame_r : ∀ E1 E2 Ef P Q,
E1 ## Ef → (P ={E1,E2}=> Q) ⊢ P ={E1 ∪ Ef,E2 ∪ Ef}=> Q).
Context (vs_frame_r : ∀ E1 E2 P Q R, (P ={E1,E2}=> Q) ⊢ P ∗ R ={E1,E2}=> Q ∗ R).
Context (vs_exists : ∀ {A} E1 E2 (Φ : A → uPred M) Q,
(∀ x, Φ x ={E1,E2}=> Q) ⊢ (∃ x, Φ x) ={E1,E2}=> Q).
Context (vs_persistent_intro_r : ∀ E1 E2 P Q R,
Persistent R →
(R -∗ (P ={E1,E2}=> Q)) ⊢ P ∗ R ={E1,E2}=> Q).
Definition fupd (E1 E2 : coPset) (P : uPred M) : uPred M :=
∃ R, R ∗ vs E1 E2 R P.
Notation "|={ E1 , E2 }=> Q" := (fupd E1 E2 Q) : bi_scope.
Global Instance fupd_ne E1 E2 : NonExpansive (@fupd E1 E2).
Proof. solve_proper. Qed.
Lemma fupd_intro E P : P ⊢ |={E,E}=> P.
Proof. iIntros "HP". iExists P. iFrame "HP". iApply vs_impl; auto. Qed.
Lemma fupd_mono E1 E2 P Q : (P ⊢ Q) → (|={E1,E2}=> P) ⊢ |={E1,E2}=> Q.
Proof.
iIntros (HPQ); iDestruct 1 as (R) "[HR Hvs]".
iExists R; iFrame "HR". iApply (vs_trans with "[$Hvs]").
iApply vs_impl. iIntros "!> HP". by iApply HPQ.
Qed.
Lemma fupd_trans E1 E2 E3 P : (|={E1,E2}=> |={E2,E3}=> P) ⊢ |={E1,E3}=> P.
Proof.
iDestruct 1 as (R) "[HR Hvs]". iExists R. iFrame "HR".
iApply (vs_trans with "[$Hvs]"). clear R.
iApply vs_exists; iIntros (R). iApply vs_persistent_intro_r; iIntros "Hvs".
iApply (vs_trans with "[$Hvs]"). iApply vs_impl; auto.
Qed.
Lemma fupd_mask_frame_r E1 E2 Ef P :
E1 ## Ef → (|={E1,E2}=> P) ⊢ |={E1 ∪ Ef,E2 ∪ Ef}=> P.
Proof.
iIntros (HE); iDestruct 1 as (R) "[HR Hvs]". iExists R; iFrame "HR".
by iApply vs_mask_frame_r.
Qed.
Lemma fupd_frame_r E1 E2 P Q : (|={E1,E2}=> P) ∗ Q ⊢ |={E1,E2}=> P ∗ Q.
Proof.
iIntros "[Hvs HQ]". iDestruct "Hvs" as (R) "[HR Hvs]".
iExists (R ∗ Q)%I. iFrame "HR HQ". by iApply vs_frame_r.
Qed.
End fupd.