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Commit 2269c6c0 authored by Ralf Jung's avatar Ralf Jung
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Merge branch 'ralf/sem-inv' into 'master'

Semantic invariants

See merge request iris/iris!319
parents a2efc56b 3db87055
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......@@ -5,6 +5,11 @@ Coq development, but not every API-breaking change is listed. Changes marked
## Iris master
**Changes in the theory of Iris itself:**
* [#] Redefine invariants as "semantic invariants" so that they support
splitting and other forms of weakening.
**Changes in Coq:**
* A new tactic `iStopProof` to turn the proof mode entailment into an ordinary
......
......@@ -6,123 +6,184 @@ From iris.base_logic.lib Require Import wsat.
Set Default Proof Using "Type".
Import uPred.
(** Derived forms and lemmas about them. *)
(** Semantic Invariants *)
Definition inv_def `{!invG Σ} (N : namespace) (P : iProp Σ) : iProp Σ :=
( i P', i (N:coPset) (P' P) ownI i P')%I.
( E, ⌜↑N E |={E,E N}=> P ( P ={E N,E}=∗ True))%I.
Definition inv_aux : seal (@inv_def). by eexists. Qed.
Definition inv {Σ i} := inv_aux.(unseal) Σ i.
Definition inv_eq : @inv = @inv_def := inv_aux.(seal_eq).
Instance: Params (@inv) 3 := {}.
Typeclasses Opaque inv.
(** * Invariants *)
Section inv.
Context `{!invG Σ}.
Implicit Types i : positive.
Implicit Types N : namespace.
Implicit Types P Q R : iProp Σ.
Global Instance inv_contractive N : Contractive (inv N).
Proof. rewrite inv_eq. solve_contractive. Qed.
Global Instance inv_ne N : NonExpansive (inv N).
Proof. apply contractive_ne, _. Qed.
Global Instance inv_proper N : Proper ((⊣⊢) ==> (⊣⊢)) (inv N).
Proof. apply ne_proper, _. Qed.
Global Instance inv_persistent N P : Persistent (inv N P).
Proof. rewrite inv_eq /inv; apply _. Qed.
Lemma inv_iff N P Q : (P Q) -∗ inv N P -∗ inv N Q.
Proof.
iIntros "#HPQ". rewrite inv_eq. iDestruct 1 as (i P') "(?&#HP&?)".
iExists i, P'. iFrame. iNext; iAlways; iSplit.
- iIntros "HP'". iApply "HPQ". by iApply "HP".
- iIntros "HQ". iApply "HP". by iApply "HPQ".
Qed.
Lemma fresh_inv_name (E : gset positive) N : i, i E i (N:coPset).
Proof.
exists (coPpick ( N gset_to_coPset E)).
rewrite -elem_of_gset_to_coPset (comm and) -elem_of_difference.
apply coPpick_elem_of=> Hfin.
eapply nclose_infinite, (difference_finite_inv _ _), Hfin.
apply gset_to_coPset_finite.
Qed.
Lemma inv_alloc N E P : P ={E}=∗ inv N P.
Proof.
rewrite inv_eq /inv_def uPred_fupd_eq. iIntros "HP [Hw $]".
iMod (ownI_alloc (. (N : coPset)) P with "[$HP $Hw]")
as (i ?) "[$ ?]"; auto using fresh_inv_name.
do 2 iModIntro. iExists i, P. rewrite -(iff_refl True%I). auto.
Qed.
Lemma inv_alloc_open N E P :
N E (|={E, E∖↑N}=> inv N P (P ={E∖↑N, E}=∗ True))%I.
Proof.
rewrite inv_eq /inv_def uPred_fupd_eq. iIntros (Sub) "[Hw HE]".
iMod (ownI_alloc_open (. (N : coPset)) P with "Hw")
as (i ?) "(Hw & #Hi & HD)"; auto using fresh_inv_name.
iAssert (ownE {[i]} ownE ( N {[i]}) ownE (E N))%I
with "[HE]" as "(HEi & HEN\i & HE\N)".
{ rewrite -?ownE_op; [|set_solver..].
rewrite assoc_L -!union_difference_L //. set_solver. }
do 2 iModIntro. iFrame "HE\N". iSplitL "Hw HEi"; first by iApply "Hw".
iSplitL "Hi".
{ iExists i, P. rewrite -(iff_refl True%I). auto. }
iIntros "HP [Hw HE\N]".
iDestruct (ownI_close with "[$Hw $Hi $HP $HD]") as "[$ HEi]".
do 2 iModIntro. iSplitL; [|done].
iCombine "HEi HEN\i HE\N" as "HEN".
rewrite -?ownE_op; [|set_solver..].
rewrite assoc_L -!union_difference_L //; set_solver.
Qed.
Lemma inv_open E N P :
N E inv N P ={E,E∖↑N}=∗ P ( P ={E∖↑N,E}=∗ True).
Proof.
rewrite inv_eq /inv_def uPred_fupd_eq /uPred_fupd_def.
iDestruct 1 as (i P') "(Hi & #HP' & #HiP)".
iDestruct "Hi" as % ?%elem_of_subseteq_singleton.
rewrite {1 4}(union_difference_L ( N) E) // ownE_op; last set_solver.
rewrite {1 5}(union_difference_L {[ i ]} ( N)) // ownE_op; last set_solver.
iIntros "(Hw & [HE $] & $) !> !>".
iDestruct (ownI_open i with "[$Hw $HE $HiP]") as "($ & HP & HD)".
iDestruct ("HP'" with "HP") as "$".
iIntros "HP [Hw $] !> !>". iApply (ownI_close _ P'). iFrame "HD Hw HiP".
iApply "HP'". iFrame.
Qed.
Lemma inv_open_strong E N P :
N E inv N P ={E,E∖↑N}=∗ P E', P ={E',N E'}=∗ True.
Proof.
iIntros (?) "Hinv".
iPoseProof (inv_open ( N) N P with "Hinv") as "H"; first done.
rewrite difference_diag_L.
iPoseProof (fupd_mask_frame_r _ _ (E N) with "H") as "H"; first set_solver.
rewrite left_id_L -union_difference_L //. iMod "H" as "[$ H]"; iModIntro.
iIntros (E') "HP".
iPoseProof (fupd_mask_frame_r _ _ E' with "(H HP)") as "H"; first set_solver.
by rewrite left_id_L.
Qed.
Global Instance into_inv_inv N P : IntoInv (inv N P) N := {}.
Global Instance into_acc_inv E N P :
IntoAcc (X:=unit) (inv N P)
(N E) True (fupd E (E∖↑N)) (fupd (E∖↑N) E)
(λ _, P)%I (λ _, P)%I (λ _, None)%I.
Proof.
rewrite /IntoAcc /accessor exist_unit.
iIntros (?) "#Hinv _". iApply inv_open; done.
Qed.
Lemma inv_open_timeless E N P `{!Timeless P} :
N E inv N P ={E,E∖↑N}=∗ P (P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?) "Hinv". iMod (inv_open with "Hinv") as "[>HP Hclose]"; auto.
iIntros "!> {$HP} HP". iApply "Hclose"; auto.
Qed.
Context `{!invG Σ}.
Implicit Types i : positive.
Implicit Types N : namespace.
Implicit Types E : coPset.
Implicit Types P Q R : iProp Σ.
(** ** Internal model of invariants *)
Definition own_inv (N : namespace) (P : iProp Σ) : iProp Σ :=
( i P', i (N:coPset) (P' P) ownI i P')%I.
Lemma own_inv_open E N P :
N E own_inv N P ={E,E∖↑N}=∗ P ( P ={E∖↑N,E}=∗ True).
Proof.
rewrite uPred_fupd_eq /uPred_fupd_def.
iDestruct 1 as (i P') "(Hi & #HP' & #HiP)".
iDestruct "Hi" as % ?%elem_of_subseteq_singleton.
rewrite {1 4}(union_difference_L ( N) E) // ownE_op; last set_solver.
rewrite {1 5}(union_difference_L {[ i ]} ( N)) // ownE_op; last set_solver.
iIntros "(Hw & [HE $] & $) !> !>".
iDestruct (ownI_open i with "[$Hw $HE $HiP]") as "($ & HP & HD)".
iDestruct ("HP'" with "HP") as "$".
iIntros "HP [Hw $] !> !>". iApply (ownI_close _ P'). iFrame "HD Hw HiP".
iApply "HP'". iFrame.
Qed.
Lemma fresh_inv_name (E : gset positive) N : i, i E i (N:coPset).
Proof.
exists (coPpick ( N gset_to_coPset E)).
rewrite -elem_of_gset_to_coPset (comm and) -elem_of_difference.
apply coPpick_elem_of=> Hfin.
eapply nclose_infinite, (difference_finite_inv _ _), Hfin.
apply gset_to_coPset_finite.
Qed.
Lemma own_inv_alloc N E P : P ={E}=∗ own_inv N P.
Proof.
rewrite uPred_fupd_eq. iIntros "HP [Hw $]".
iMod (ownI_alloc (. (N : coPset)) P with "[$HP $Hw]")
as (i ?) "[$ ?]"; auto using fresh_inv_name.
do 2 iModIntro. iExists i, P. rewrite -(iff_refl True%I). auto.
Qed.
Lemma own_inv_alloc_open N E P :
N E (|={E, E∖↑N}=> own_inv N P (P ={E∖↑N, E}=∗ True))%I.
Proof.
rewrite uPred_fupd_eq. iIntros (Sub) "[Hw HE]".
iMod (ownI_alloc_open (. (N : coPset)) P with "Hw")
as (i ?) "(Hw & #Hi & HD)"; auto using fresh_inv_name.
iAssert (ownE {[i]} ownE ( N {[i]}) ownE (E N))%I
with "[HE]" as "(HEi & HEN\i & HE\N)".
{ rewrite -?ownE_op; [|set_solver..].
rewrite assoc_L -!union_difference_L //. set_solver. }
do 2 iModIntro. iFrame "HE\N". iSplitL "Hw HEi"; first by iApply "Hw".
iSplitL "Hi".
{ iExists i, P. rewrite -(iff_refl True%I). auto. }
iIntros "HP [Hw HE\N]".
iDestruct (ownI_close with "[$Hw $Hi $HP $HD]") as "[$ HEi]".
do 2 iModIntro. iSplitL; [|done].
iCombine "HEi HEN\i HE\N" as "HEN".
rewrite -?ownE_op; [|set_solver..].
rewrite assoc_L -!union_difference_L //; set_solver.
Qed.
Lemma own_inv_to_inv M P: own_inv M P -∗ inv M P.
Proof.
iIntros "#I". rewrite inv_eq. iIntros (E H).
iPoseProof (own_inv_open with "I") as "H"; eauto.
Qed.
(** ** Public API of invariants *)
Global Instance inv_contractive N : Contractive (inv N).
Proof. rewrite inv_eq. solve_contractive. Qed.
Global Instance inv_ne N : NonExpansive (inv N).
Proof. apply contractive_ne, _. Qed.
Global Instance inv_proper N : Proper (equiv ==> equiv) (inv N).
Proof. apply ne_proper, _. Qed.
Global Instance inv_persistent N P : Persistent (inv N P).
Proof. rewrite inv_eq. apply _. Qed.
Lemma inv_acc N P Q:
inv N P -∗ (P -∗ Q (Q -∗ P)) -∗ inv N Q.
Proof.
rewrite inv_eq. iIntros "#HI #Acc !>" (E H).
iMod ("HI" $! E H) as "[HP Hclose]".
iDestruct ("Acc" with "HP") as "[$ HQP]".
iIntros "!> HQ". iApply "Hclose". iApply "HQP". done.
Qed.
Lemma inv_iff N P Q : (P Q) -∗ inv N P -∗ inv N Q.
Proof.
iIntros "#HPQ #HI". iApply (inv_acc with "HI").
iIntros "!> !# HP". iSplitL "HP".
- by iApply "HPQ".
- iIntros "HQ". by iApply "HPQ".
Qed.
Lemma inv_alloc N E P : P ={E}=∗ inv N P.
Proof.
iIntros "HP". iApply own_inv_to_inv.
iApply (own_inv_alloc N E with "HP").
Qed.
Lemma inv_alloc_open N E P :
N E (|={E, E∖↑N}=> inv N P (P ={E∖↑N, E}=∗ True))%I.
Proof.
iIntros (?). iMod own_inv_alloc_open as "[HI $]"; first done.
iApply own_inv_to_inv. done.
Qed.
Lemma inv_open E N P :
N E inv N P ={E,E∖↑N}=∗ P ( P ={E∖↑N,E}=∗ True).
Proof.
rewrite inv_eq /inv_def; iIntros (?) "#HI". by iApply "HI".
Qed.
(** ** Proof mode integration *)
Global Instance into_inv_inv N P : IntoInv (inv N P) N := {}.
Global Instance into_acc_inv N P E:
IntoAcc (X := unit) (inv N P)
(N E) True (fupd E (E N)) (fupd (E N) E)
(λ _ : (), ( P)%I) (λ _ : (), ( P)%I) (λ _ : (), None).
Proof.
rewrite inv_eq /IntoAcc /accessor bi.exist_unit.
iIntros (?) "#Hinv _". iApply "Hinv"; done.
Qed.
(** ** Derived properties *)
Lemma inv_open_strong E N P :
N E inv N P ={E,E∖↑N}=∗ P E', P ={E',N E'}=∗ True.
Proof.
iIntros (?) "Hinv".
iPoseProof (inv_open ( N) N P with "Hinv") as "H"; first done.
rewrite difference_diag_L.
iPoseProof (fupd_mask_frame_r _ _ (E N) with "H") as "H"; first set_solver.
rewrite left_id_L -union_difference_L //. iMod "H" as "[$ H]"; iModIntro.
iIntros (E') "HP".
iPoseProof (fupd_mask_frame_r _ _ E' with "(H HP)") as "H"; first set_solver.
by rewrite left_id_L.
Qed.
Lemma inv_open_timeless E N P `{!Timeless P} :
N E inv N P ={E,E∖↑N}=∗ P (P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?) "Hinv". iMod (inv_open with "Hinv") as "[>HP Hclose]"; auto.
iIntros "!> {$HP} HP". iApply "Hclose"; auto.
Qed.
Lemma inv_sep_l N P Q : inv N (P Q) -∗ inv N P.
Proof.
iIntros "#HI". iApply inv_acc; eauto.
iIntros "!> !# [$ $] $".
Qed.
Lemma inv_sep_r N P Q : inv N (P Q) -∗ inv N Q.
Proof.
rewrite (comm _ P Q). eapply inv_sep_l.
Qed.
Lemma inv_sep N P Q : inv N (P Q) -∗ inv N P inv N Q.
Proof.
iIntros "#H".
iPoseProof (inv_sep_l with "H") as "$".
iPoseProof (inv_sep_r with "H") as "$".
Qed.
End inv.
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