Commit a180e106 by 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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