Show that the fancy update can be encoded in terms of the view shift.

_CoqProject

@@ -51,6 +51,7 @@ theories/base_logic/lib/counter_examples.v
 theories/base_logic/lib/fractional.v
 theories/base_logic/lib/gen_heap.v
 theories/base_logic/lib/core.v
+theories/base_logic/lib/fancy_updates_from_vs.v
 theories/program_logic/adequacy.v
 theories/program_logic/lifting.v
 theories/program_logic/weakestpre.v

theories/base_logic/lib/fancy_updates_from_vs.v

+(* This file shows that the fancy update can be encoded in terms of the
+view shift, and that the laws of the fancy update can be derived from the
+laws of the view shift. *)
+From iris.proofmode Require Import tactics.
+From stdpp Require Export coPset.
+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") : uPred_scope.
+
+Context (vs_ne : ∀ E1 E2, NonExpansive2 (vs E1 E2)).
+Context (vs_persistent : ∀ E1 E2 P Q, PersistentP (P ={E1,E2}=> Q)).
+
+Context (vs_impl : ∀ E P Q, □ (P → Q) ⊢ P ={E,E}=> Q).
+Context (vs_transitive : ∀ 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,
+  PersistentP 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)%I.
+
+Notation "|={ E1 , E2 }=> Q" := (fupd E1 E2 Q)
+  (at level 99, E1, E2 at level 50, Q at level 200,
+   format "|={ E1 , E2 }=>  Q") : uPred_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_transitive with "[$Hvs]").
+  iApply vs_impl. iIntros "!# HP". by iApply HPQ.
+Qed.
+
+Lemma fupd_trans E1 E2 E3 P : (|={E1,E2}=> |={E2,E3}=> P) ⊢ |={E1,E3}=> P.
+Proof.
+  iDestruct 1 as (R) "[HR Hvs]". iExists R. iFrame "HR".
+  iApply (vs_transitive with "[$Hvs]"). clear R.
+  iApply vs_exists; iIntros (R). iApply vs_persistent_intro_r; iIntros "Hvs".
+  iApply (vs_transitive 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.
\ No newline at end of file