cancelable_invariants.v 3.89 KB
 Robbert Krebbers committed Oct 30, 2017 1 ``````From iris.base_logic.lib Require Export invariants. `````` Ralf Jung committed Feb 15, 2018 2 ``````From iris.bi.lib Require Import fractional. `````` Robbert Krebbers committed Aug 25, 2016 3 ``````From iris.algebra Require Export frac. `````` Robbert Krebbers committed Oct 05, 2016 4 ``````From iris.proofmode Require Import tactics. `````` Ralf Jung committed Jan 05, 2017 5 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Aug 25, 2016 6 7 8 ``````Import uPred. Class cinvG Σ := cinv_inG :> inG Σ fracR. `````` Ralf Jung committed Apr 07, 2017 9 10 11 12 ``````Definition cinvΣ : gFunctors := #[GFunctor fracR]. Instance subG_cinvΣ {Σ} : subG cinvΣ Σ → cinvG Σ. Proof. solve_inG. Qed. `````` Robbert Krebbers committed Aug 25, 2016 13 14 `````` Section defs. `````` Robbert Krebbers committed Oct 28, 2016 15 `````` Context `{invG Σ, cinvG Σ}. `````` Robbert Krebbers committed Aug 25, 2016 16 17 18 19 `````` Definition cinv_own (γ : gname) (p : frac) : iProp Σ := own γ p. Definition cinv (N : namespace) (γ : gname) (P : iProp Σ) : iProp Σ := `````` Ralf Jung committed Apr 11, 2017 20 `````` (∃ P', □ ▷ (P ↔ P') ∗ inv N (P' ∨ cinv_own γ 1%Qp))%I. `````` Robbert Krebbers committed Aug 25, 2016 21 22 ``````End defs. `````` Robbert Krebbers committed Oct 28, 2016 23 ``````Instance: Params (@cinv) 5. `````` Robbert Krebbers committed Aug 25, 2016 24 25 `````` Section proofs. `````` Robbert Krebbers committed Oct 28, 2016 26 `````` Context `{invG Σ, cinvG Σ}. `````` Robbert Krebbers committed Aug 25, 2016 27 `````` `````` Robbert Krebbers committed Oct 25, 2017 28 `````` Global Instance cinv_own_timeless γ p : Timeless (cinv_own γ p). `````` Robbert Krebbers committed Aug 25, 2016 29 30 `````` Proof. rewrite /cinv_own; apply _. Qed. `````` Ralf Jung committed Apr 11, 2017 31 32 `````` Global Instance cinv_contractive N γ : Contractive (cinv N γ). Proof. solve_contractive. Qed. `````` Ralf Jung committed Jan 27, 2017 33 `````` Global Instance cinv_ne N γ : NonExpansive (cinv N γ). `````` Ralf Jung committed Apr 11, 2017 34 `````` Proof. exact: contractive_ne. Qed. `````` Robbert Krebbers committed Aug 25, 2016 35 `````` Global Instance cinv_proper N γ : Proper ((≡) ==> (≡)) (cinv N γ). `````` Ralf Jung committed Apr 11, 2017 36 `````` Proof. exact: ne_proper. Qed. `````` Robbert Krebbers committed Aug 25, 2016 37 `````` `````` Robbert Krebbers committed Oct 25, 2017 38 `````` Global Instance cinv_persistent N γ P : Persistent (cinv N γ P). `````` Robbert Krebbers committed Aug 25, 2016 39 40 `````` Proof. rewrite /cinv; apply _. Qed. `````` Robbert Krebbers committed Mar 28, 2018 41 `````` Global Instance cinv_own_fractional γ : Fractional (cinv_own γ). `````` Robbert Krebbers committed Oct 30, 2017 42 `````` Proof. intros ??. by rewrite /cinv_own -own_op. Qed. `````` Robbert Krebbers committed Mar 28, 2018 43 `````` Global Instance cinv_own_as_fractional γ q : `````` Jacques-Henri Jourdan committed Nov 23, 2016 44 `````` AsFractional (cinv_own γ q) (cinv_own γ) q. `````` Jacques-Henri Jourdan committed Dec 13, 2016 45 `````` Proof. split. done. apply _. Qed. `````` Robbert Krebbers committed Aug 25, 2016 46 `````` `````` Robbert Krebbers committed Dec 09, 2016 47 48 `````` Lemma cinv_own_valid γ q1 q2 : cinv_own γ q1 -∗ cinv_own γ q2 -∗ ✓ (q1 + q2)%Qp. Proof. apply (own_valid_2 γ q1 q2). Qed. `````` Robbert Krebbers committed Aug 25, 2016 49 `````` `````` Robbert Krebbers committed Dec 09, 2016 50 51 52 53 54 `````` Lemma cinv_own_1_l γ q : cinv_own γ 1 -∗ cinv_own γ q -∗ False. Proof. iIntros "H1 H2". iDestruct (cinv_own_valid with "H1 H2") as %[]%(exclusive_l 1%Qp). Qed. `````` Robbert Krebbers committed Aug 25, 2016 55 `````` `````` Ralf Jung committed Apr 11, 2017 56 57 58 59 `````` Lemma cinv_iff N γ P P' : ▷ □ (P ↔ P') -∗ cinv N γ P -∗ cinv N γ P'. Proof. iIntros "#HP' Hinv". iDestruct "Hinv" as (P'') "[#HP'' Hinv]". `````` Robbert Krebbers committed Mar 03, 2018 60 `````` iExists _. iFrame "Hinv". iAlways. iNext. iSplit. `````` Ralf Jung committed Apr 11, 2017 61 62 63 64 `````` - iIntros "?". iApply "HP''". iApply "HP'". done. - iIntros "?". iApply "HP'". iApply "HP''". done. Qed. `````` Robbert Krebbers committed Nov 03, 2016 65 `````` Lemma cinv_alloc E N P : ▷ P ={E}=∗ ∃ γ, cinv N γ P ∗ cinv_own γ 1. `````` Robbert Krebbers committed Aug 25, 2016 66 `````` Proof. `````` Ralf Jung committed Apr 11, 2017 67 `````` iIntros "HP". `````` Robbert Krebbers committed Oct 25, 2016 68 `````` iMod (own_alloc 1%Qp) as (γ) "H1"; first done. `````` Ralf Jung committed Apr 11, 2017 69 70 `````` iMod (inv_alloc N _ (P ∨ own γ 1%Qp)%I with "[HP]"); first by eauto. iExists _. iFrame. iExists _. iFrame. iIntros "!> !# !>". iSplit; by iIntros "?". `````` Robbert Krebbers committed Aug 25, 2016 71 72 `````` Qed. `````` 73 `````` Lemma cinv_cancel E N γ P : ↑N ⊆ E → cinv N γ P -∗ cinv_own γ 1 ={E}=∗ ▷ P. `````` Robbert Krebbers committed Aug 25, 2016 74 `````` Proof. `````` Ralf Jung committed Apr 11, 2017 75 76 77 78 79 `````` iIntros (?) "#Hinv Hγ". iDestruct "Hinv" as (P') "[#HP' Hinv]". iInv N as "[HP|>Hγ']" "Hclose". - iMod ("Hclose" with "[Hγ]") as "_"; first by eauto. iModIntro. iNext. iApply "HP'". done. - iDestruct (cinv_own_1_l with "Hγ Hγ'") as %[]. `````` Robbert Krebbers committed Aug 25, 2016 80 81 82 `````` Qed. Lemma cinv_open E N γ p P : `````` Robbert Krebbers committed Nov 22, 2016 83 `````` ↑N ⊆ E → `````` 84 `````` cinv N γ P -∗ cinv_own γ p ={E,E∖↑N}=∗ ▷ P ∗ cinv_own γ p ∗ (▷ P ={E∖↑N,E}=∗ True). `````` Robbert Krebbers committed Aug 25, 2016 85 `````` Proof. `````` Ralf Jung committed Apr 11, 2017 86 87 88 89 90 `````` iIntros (?) "#Hinv Hγ". iDestruct "Hinv" as (P') "[#HP' Hinv]". iInv N as "[HP | >Hγ']" "Hclose". - iIntros "!> {\$Hγ}". iSplitL "HP". + iNext. iApply "HP'". done. + iIntros "HP". iApply "Hclose". iLeft. iNext. by iApply "HP'". `````` Robbert Krebbers committed Dec 09, 2016 91 `````` - iDestruct (cinv_own_1_l with "Hγ' Hγ") as %[]. `````` Robbert Krebbers committed Aug 25, 2016 92 `````` Qed. `````` Joseph Tassarotti committed Feb 23, 2018 93 94 `````` Global Instance into_inv_cinv N γ P : IntoInv (cinv N γ P) N. `````` Robbert Krebbers committed Feb 23, 2018 95 `````` Global Instance elim_inv_cinv p γ E N P Q Q' : `````` Robbert Krebbers committed Apr 04, 2018 96 `````` (∀ R, ElimModal True false false (|={E,E∖↑N}=> R) R Q Q') → `````` Robbert Krebbers committed Feb 23, 2018 97 98 `````` ElimInv (↑N ⊆ E) (cinv N γ P) (cinv_own γ p) (▷ P ∗ cinv_own γ p) (▷ P ={E∖↑N,E}=∗ True) Q Q'. `````` Joseph Tassarotti committed Feb 23, 2018 99 `````` Proof. `````` Robbert Krebbers committed Feb 23, 2018 100 `````` rewrite /ElimInv /ElimModal. iIntros (Helim ?) "(#H1&Hown&H2)". `````` Robbert Krebbers committed Apr 04, 2018 101 `````` iApply Helim; [done|]; simpl. iSplitR "H2"; [|done]. `````` Joseph Tassarotti committed Feb 23, 2018 102 103 104 `````` iMod (cinv_open E N γ p P with "[#] [Hown]") as "(HP&Hown&Hclose)"; auto. by iFrame. Qed. `````` Robbert Krebbers committed Aug 25, 2016 105 ``````End proofs. `````` Ralf Jung committed Apr 11, 2017 106 107 `````` Typeclasses Opaque cinv_own cinv.``````