Commit a180e106 authored by Robbert Krebbers's avatar Robbert Krebbers

Close invariant under `iProto_le`. Remove updates from `iProto_le`.

Symmetric proofs admitted.
parent 0010b710
......@@ -204,11 +204,11 @@ Program Definition iProto_le_aux `{invG Σ} (rec : iProto Σ -n> iProto Σ -n> i
((p1 proto_end p2 proto_end)
( pc1 pc2,
p1 proto_message Send pc1 p2 proto_message Send pc2
v p2', pc2 v (proto_eq_next p2') ={}=
v p2', pc2 v (proto_eq_next p2') -
p1', rec p1' p2' pc1 v (proto_eq_next p1'))
( pc1 pc2,
p1 proto_message Receive pc1 p2 proto_message Receive pc2
v p1', pc1 v (proto_eq_next p1') ={}=
v p1', pc1 v (proto_eq_next p1') -
p2', rec p1' p2' pc2 v (proto_eq_next p2')))%I.
Solve Obligations with solve_proper.
Local Instance iProto_le_aux_contractive `{invG Σ} : Contractive (@iProto_le_aux Σ _).
......@@ -225,7 +225,7 @@ Fixpoint proto_interp {Σ} (vs : list val) (p1 p2 : iProto Σ) : iProp Σ :=
p2 proto_message Receive pc
pc v (proto_eq_next p2')
proto_interp vs p1 p2'
end%I.
end.
Arguments proto_interp {_} _ _%proto _%proto : simpl nomatch.
Record proto_name := ProtName {
......@@ -246,26 +246,28 @@ Definition proto_own_auth `{!proto_chanG Σ} (γ : proto_name) (s : side)
(p : iProto Σ) : iProp Σ :=
own (side_elim s proto_l_name proto_r_name γ) (E (Next (to_pre_proto p))).
Definition proto_inv `{!proto_chanG Σ} (γ : proto_name) : iProp Σ :=
( vsl vsr pl pr,
Definition proto_inv `{!invG Σ, proto_chanG Σ} (γ : proto_name) : iProp Σ :=
vsl vsr pl pr pl' pr',
chan_own (proto_c_name γ) Left vsl
chan_own (proto_c_name γ) Right vsr
proto_own_auth γ Left pl
proto_own_auth γ Right pr
((vsr = [] proto_interp vsl pl pr)
(vsl = [] proto_interp vsr pr pl)))%I.
proto_own_auth γ Left pl'
proto_own_auth γ Right pr'
(iProto_le pl pl'
iProto_le pr pr'
((vsr = [] proto_interp vsl pl pr)
(vsl = [] proto_interp vsr pr pl))).
Definition protoN := nroot .@ "proto".
(** * The connective for ownership of channel ends *)
Definition mapsto_proto_def `{!proto_chanG Σ, !heapG Σ}
(c : val) (p : iProto Σ) : iProp Σ :=
( s (c1 c2 : val) γ p',
s (c1 c2 : val) γ p',
c = side_elim s c1 c2
iProto_le p' p
proto_own_frag γ s p'
is_chan protoN (proto_c_name γ) c1 c2
inv protoN (proto_inv γ))%I.
inv protoN (proto_inv γ).
Definition mapsto_proto_aux : seal (@mapsto_proto_def). by eexists. Qed.
Definition mapsto_proto {Σ pΣ hΣ} := mapsto_proto_aux.(unseal) Σ pΣ hΣ.
Definition mapsto_proto_eq :
......@@ -422,9 +424,9 @@ Section proto.
iLöb as "IH" forall (p). destruct (proto_case p) as [->|([]&pc&->)].
- rewrite iProto_le_unfold. iLeft; by auto.
- rewrite iProto_le_unfold. iRight; iLeft. iExists _, _; do 2 (iSplit; [done|]).
iIntros (v p') "Hpc". iExists p'. iIntros "{$Hpc} !> !>". iApply "IH".
iIntros (v p') "Hpc". iExists p'. iIntros "{$Hpc} !>". iApply "IH".
- rewrite iProto_le_unfold. iRight; iRight. iExists _, _; do 2 (iSplit; [done|]).
iIntros (v p') "Hpc". iExists p'. iIntros "{$Hpc} !> !>". iApply "IH".
iIntros (v p') "Hpc". iExists p'. iIntros "{$Hpc} !>". iApply "IH".
Qed.
Lemma iProto_le_end_inv p : iProto_le p proto_end - p proto_end.
......@@ -437,7 +439,7 @@ Section proto.
Lemma iProto_le_send_inv p1 pc2 :
iProto_le p1 (proto_message Send pc2) - pc1,
p1 proto_message Send pc1
v p2', pc2 v (proto_eq_next p2') ={}=
v p2', pc2 v (proto_eq_next p2') -
p1', iProto_le p1' p2' pc1 v (proto_eq_next p1').
Proof.
rewrite iProto_le_unfold. iIntros "[[_ Heq]|[H|H]]".
......@@ -454,7 +456,7 @@ Section proto.
Lemma iProto_le_recv_inv p1 pc2 :
iProto_le p1 (proto_message Receive pc2) - pc1,
p1 proto_message Receive pc1
v p1', pc1 v (proto_eq_next p1') ={}=
v p1', pc1 v (proto_eq_next p1') -
p2', iProto_le p1' p2' pc2 v (proto_eq_next p2').
Proof.
rewrite iProto_le_unfold. iIntros "[[_ Heq]|[H|H]]".
......@@ -465,7 +467,7 @@ Section proto.
iDestruct (proto_message_equivI with "Heq") as "[_ #Heq]".
iExists pc1. iIntros "{$Hp1}" (v p1') "Hpc".
iSpecialize ("Heq" $! v). iDestruct (bi.ofe_morO_equivI with "Heq") as "Heq'".
iMod ("H" with "Hpc") as (p2') "[Hle Hpc]". iModIntro.
iDestruct ("H" with "Hpc") as (p2') "[Hle Hpc]".
iExists p2'. iFrame "Hle". by iRewrite ("Heq'" $! (proto_eq_next p2')).
Qed.
......@@ -481,8 +483,8 @@ Section proto.
iRewrite "Hp1". rewrite iProto_le_unfold; iRight; iLeft.
iExists _, _; do 2 (iSplit; [done|]).
iIntros (v p3') "Hpc".
iMod ("H3" with "Hpc") as (p2') "[Hle Hpc]".
iMod ("H2" with "Hpc") as (p1') "[Hle' Hpc]".
iDestruct ("H3" with "Hpc") as (p2') "[Hle Hpc]".
iDestruct ("H2" with "Hpc") as (p1') "[Hle' Hpc]".
iExists p1'. iIntros "{$Hpc} !>". by iApply ("IH" with "Hle'").
- iDestruct (iProto_le_recv_inv with "H2") as (pc2) "[Hp2 H3]".
iRewrite "Hp2" in "H1".
......@@ -490,14 +492,14 @@ Section proto.
iRewrite "Hp1". rewrite iProto_le_unfold; iRight; iRight.
iExists _, _; do 2 (iSplit; [done|]).
iIntros (v p1') "Hpc".
iMod ("H2" with "Hpc") as (p2') "[Hle Hpc]".
iMod ("H3" with "Hpc") as (p3') "[Hle' Hpc]".
iDestruct ("H2" with "Hpc") as (p2') "[Hle Hpc]".
iDestruct ("H3" with "Hpc") as (p3') "[Hle' Hpc]".
iExists p3'. iIntros "{$Hpc} !>". by iApply ("IH" with "Hle").
Qed.
Lemma iProto_send_le {TT1 TT2} (pc1 : TT1 val * iProp Σ * iProto Σ)
(pc2 : TT2 val * iProp Σ * iProto Σ) :
(.. x2 : TT2, (pc2 x2).1.2 ={}= .. x1 : TT1,
(.. x2 : TT2, (pc2 x2).1.2 - .. x1 : TT1,
(pc1 x1).1.1 = (pc2 x2).1.1
(pc1 x1).1.2
iProto_le (pc1 x1).2 (pc2 x2).2) -
......@@ -506,16 +508,16 @@ Section proto.
iIntros "H". rewrite iProto_le_unfold iProto_message_eq. iRight; iLeft.
iExists _, _; do 2 (iSplit; [done|]).
iIntros (v p2') "/=". iDestruct 1 as (x2 ->) "[Hpc #Heq]".
iMod ("H" with "Hpc") as (x1 ?) "[Hpc Hle]".
iDestruct ("H" with "Hpc") as (x1 ?) "[Hpc Hle]".
iExists (pc1 x1).2. iSplitL "Hle".
{ iIntros "!> !>". change (fixpoint iProto_le_aux ?p1 ?p2) with (iProto_le p1 p2).
{ iIntros "!>". change (fixpoint iProto_le_aux ?p1 ?p2) with (iProto_le p1 p2).
by iRewrite "Heq". }
iModIntro. iExists x1. iSplit; [done|]. iSplit; [by iApply "Hpc"|done].
iExists x1. iSplit; [done|]. iSplit; [by iApply "Hpc"|done].
Qed.
Lemma iProto_recv_le {TT1 TT2} (pc1 : TT1 val * iProp Σ * iProto Σ)
(pc2 : TT2 val * iProp Σ * iProto Σ) :
(.. x1 : TT1, (pc1 x1).1.2 ={}= .. x2 : TT2,
(.. x1 : TT1, (pc1 x1).1.2 - .. x2 : TT2,
(pc1 x1).1.1 = (pc2 x2).1.1
(pc2 x2).1.2
iProto_le (pc1 x1).2 (pc2 x2).2) -
......@@ -524,10 +526,10 @@ Section proto.
iIntros "H". rewrite iProto_le_unfold iProto_message_eq. iRight; iRight.
iExists _, _; do 2 (iSplit; [done|]).
iIntros (v p1') "/=". iDestruct 1 as (x1 ->) "[Hpc #Heq]".
iMod ("H" with "Hpc") as (x2 ?) "[Hpc Hle]". iExists (pc2 x2).2. iSplitL "Hle".
{ iIntros "!> !>". change (fixpoint iProto_le_aux ?p1 ?p2) with (iProto_le p1 p2).
iDestruct ("H" with "Hpc") as (x2 ?) "[Hpc Hle]". iExists (pc2 x2).2. iSplitL "Hle".
{ iIntros "!>". change (fixpoint iProto_le_aux ?p1 ?p2) with (iProto_le p1 p2).
by iRewrite "Heq". }
iModIntro. iExists x2. iSplit; [done|]. iSplit; [by iApply "Hpc"|done].
iExists x2. iSplit; [done|]. iSplit; [by iApply "Hpc"|done].
Qed.
Lemma iProto_mapsto_le c p1 p2 : c p1 - iProto_le p1 p2 - c p2.
......@@ -640,7 +642,8 @@ Section proto.
{ by apply excl_auth_valid. }
pose (ProtName cγ lγ rγ) as pγ.
iMod (inv_alloc protoN _ (proto_inv pγ) with "[-Hlstf Hrstf Hcctx]") as "#Hinv".
{ iNext. rewrite /proto_inv. eauto 10 with iFrame. }
{ iNext. iExists [], [], p, (iProto_dual p), p, (iProto_dual p). iFrame.
do 2 (iSplitL; [iApply iProto_le_refl|]). auto. }
iModIntro. rewrite mapsto_proto_eq. iSplitL "Hlstf".
- iExists Left, c1, c2, pγ, p.
iFrame "Hlstf Hinv Hcctx". iSplit; [done|]. iApply iProto_le_refl.
......@@ -665,26 +668,32 @@ Section proto.
iExists s, c1, c2, γ. iSplit; first done. iFrame "Hcctx".
iDestruct (iProto_le_send_inv with "Hle") as (pc') "[Hp H] /=".
iRewrite "Hp" in "Hst"; clear p.
iMod ("H" with "[HP]") as (p1') "[Hle HP]".
iDestruct ("H" with "[HP]") as (p1') "[Hle HP]".
{ iExists _. iFrame "HP". by iSplit. }
iInv protoN as (l r pl pr) "(>Hcl & >Hcr & Hstla & Hstra & Hinv')" "Hclose".
iInv protoN as (vsl vsr pl pr pl' pr')
"(>Hcl & >Hcr & Hstla & Hstra & Hlel & Hler & Hinv')" "Hclose".
(* TODO: refactor to avoid twice nearly the same proof *)
iModIntro. destruct s.
- iExists _. iIntros "{$Hcl} !> Hcl".
iDestruct (proto_own_auth_agree with "Hstla Hst") as "#Heq".
iMod (proto_own_auth_update _ _ _ _ p1' with "Hstla Hst") as "[Hstla Hst]".
iMod ("Hclose" with "[-Hst Hle]") as "_".
{ iNext. iExists _,_,_,_. iFrame "Hcr Hstra Hstla Hcl". iLeft.
iRewrite "Heq" in "Hinv'".
iDestruct "Hinv'" as "[[-> Heval]|[-> Heval]]".
{ iSplit=> //. by iApply (proto_interp_send with "Heval [HP]"). }
destruct r as [|vr r]; last first.
iMod (proto_own_auth_update _ _ _ _ (pc x).2 with "Hstla Hst") as "[Hstla Hst]".
iMod ("Hclose" with "[-Hst]") as "_".
{ iNext. iRewrite "Heq" in "Hlel". iClear (pl') "Heq".
iDestruct (iProto_le_send_inv with "Hlel") as (pc'') "[Hpl H] /=".
iRewrite "Hpl" in "Hinv'"; clear pl.
iDestruct ("H" with "HP") as (p1'') "[Hlel HP]".
iExists _, _, _, _, _, _. iFrame "Hcr Hstra Hstla Hcl Hler".
iNext. iSplitL "Hle Hlel".
{ by iApply (iProto_le_trans with "[$]"). }
iLeft. iDestruct "Hinv'" as "[[-> Heval]|[-> Heval]]".
{ iSplit=> //. iApply (proto_interp_send with "Heval HP"). }
destruct vsr as [|vr vsr]; last first.
{ iDestruct (proto_interp_False with "Heval") as %[]. }
iSplit; first done; simpl. iRewrite (proto_interp_nil with "Heval").
iApply (proto_interp_send _ [] with "[//] HP"). }
iModIntro. rewrite mapsto_proto_eq. iExists Left, c1, c2, γ, p1'.
by iFrame "Hcctx Hinv Hst Hle".
- iExists _. iIntros "{$Hcr} !> Hcr".
iModIntro. rewrite mapsto_proto_eq. iExists Left, c1, c2, γ, (pc x).2.
iFrame "Hcctx Hinv Hst". iSplit; [done|]. iApply iProto_le_refl.
- (* iExists _. iIntros "{$Hcr} !> Hcr".
iDestruct (proto_own_auth_agree with "Hstra Hst") as "#Heq".
iMod (proto_own_auth_update _ _ _ _ p1' with "Hstra Hst") as "[Hstra Hst]".
iMod ("Hclose" with "[-Hst Hle]") as "_".
......@@ -697,8 +706,8 @@ Section proto.
iSplit; first done; simpl. iRewrite (proto_interp_nil with "Heval").
iApply (proto_interp_send _ [] with "[//] HP"). }
iModIntro. rewrite mapsto_proto_eq. iExists Right, c1, c2, γ, p1'.
by iFrame "Hcctx Hinv Hst Hle".
Qed.
by iFrame "Hcctx Hinv Hst Hle". *) admit.
Admitted.
Lemma proto_recv_acc {TT} c (pc : TT val * iProp Σ * iProto Σ) :
c iProto_message Receive pc -
......@@ -719,34 +728,42 @@ Section proto.
iRewrite "Hp" in "Hst". clear p.
iExists (side_elim s Right Left), c1, c2, γ. iSplit; first by destruct s.
iFrame "Hcctx".
iInv protoN as (l r pl pr) "(>Hcl & >Hcr & Hstla & Hstra & Hinv')" "Hclose".
iExists (side_elim s r l). iModIntro.
iInv protoN as (vsl vsr pl pr pl' pr')
"(>Hcl & >Hcr & Hstla & Hstra & Hlel & Hler & Hinv')" "Hclose".
iExists (side_elim s vsr vsl). iModIntro.
(* TODO: refactor to avoid twice nearly the same proof *)
destruct s; simpl.
- iIntros "{$Hcr} !>".
iDestruct (proto_own_auth_agree with "Hstla Hst") as "#Heq".
iDestruct (proto_own_auth_agree with "Hstla Hst") as "#Hpl'".
iSplit.
+ iIntros "Hcr".
iMod ("Hclose" with "[-Hst Hle]") as "_".
{ iNext. iExists l, r, _, _. iFrame. }
{ iNext. iExists vsl, vsr, _, _, _, _. iFrame. }
iModIntro. rewrite mapsto_proto_eq.
iExists Left, c1, c2, γ, (proto_message Receive pc').
iFrame "Hcctx Hinv Hst". iSplit; first done.
rewrite iProto_le_unfold. iRight; auto 10.
+ iIntros (v vs ->) "Hcr".
iDestruct "Hinv'" as "[[% _]|[-> Heval]]"; first done.
iAssert ( proto_interp (v :: vs) pr (proto_message Receive pc'))%I
iDestruct "Hinv'" as "[[>% _]|[>-> Heval]]"; first done.
iAssert ( iProto_le pl (proto_message Receive pc'))%I with "[Hlel]" as "Hlel".
{ iNext. by iRewrite "Hpl'" in "Hlel". }
iDestruct (iProto_le_recv_inv with "Hlel") as (pc'') "[#Hpl Hlel] /=".
iAssert ( proto_interp (v :: vs) pr (proto_message Receive pc''))%I
with "[Heval]" as "Heval".
{ iNext. by iRewrite "Heq" in "Heval". }
{ iNext. by iRewrite "Hpl" in "Heval". }
iDestruct (proto_interp_recv with "Heval") as (q) "[Hpc Heval]".
iMod (proto_own_auth_update _ _ _ _ q with "Hstla Hst") as "[Hstla Hst]".
iMod ("Hclose" with "[-Hst Hpc Hle]") as %_.
{ iExists _, _,_ ,_; iFrame; eauto. }
iIntros "!> !>". iMod ("Hle" with "Hpc") as (q') "[Hle H]".
iDestruct "H" as (x) "(Hv & HP & #Hf) /=".
iIntros "!> !>". iExists x. iFrame "Hv HP". iRewrite -"Hf".
rewrite mapsto_proto_eq. iExists Left, c1, c2, γ, q. iFrame; auto.
- iIntros "{$Hcl} !>".
iDestruct ("Hlel" with "Hpc") as (p1'') "[Hlel Hpc]".
iDestruct ("Hle" with "Hpc") as (p1''') "[Hle Hpc]".
iMod (proto_own_auth_update _ _ _ _ p1''' with "Hstla Hst") as "[Hstla Hst]".
iMod ("Hclose" with "[-Hst Hpc]") as %_.
{ iExists _, _, q, _, _, _. iFrame "Hcl Hcr Hstra Hstla Hler".
iIntros "!> !>". iSplitL "Hle Hlel"; last by auto.
by iApply (iProto_le_trans with "[$]"). }
iIntros "!> !> !>". iDestruct "Hpc" as (x) "(Hv & HP & #Hf) /=".
iIntros "!>". iExists x. iFrame "Hv HP". iRewrite -"Hf".
rewrite mapsto_proto_eq. iExists Left, c1, c2, γ, p1'''.
iFrame "Hst Hcctx Hinv". iSplit; [done|]. iApply iProto_le_refl.
- (* iIntros "{$Hcl} !>".
iDestruct (proto_own_auth_agree with "Hstra Hst") as "#Heq".
iSplit.
+ iIntros "Hcl".
......@@ -769,7 +786,7 @@ Section proto.
iDestruct "H" as (x) "(Hv & HP & Hf) /=".
iIntros "!> !>". iExists x. iFrame "Hv HP". iRewrite -"Hf".
rewrite mapsto_proto_eq. iExists Right, c1, c2, γ, q. iFrame; auto.
Qed.
Qed. *) Admitted.
(** ** Specifications of [send] and [recv] *)
Lemma new_chan_proto_spec :
......
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