### Splitting and combining boxes. Also refactored boxes by currying things.

parent ff96075a
 ... @@ -49,8 +49,12 @@ Global Instance box_inv_ne n γ : Proper (dist n ==> dist n) (slice_inv γ). ... @@ -49,8 +49,12 @@ Global Instance box_inv_ne n γ : Proper (dist n ==> dist n) (slice_inv γ). Proof. solve_proper. Qed. Proof. solve_proper. Qed. Global Instance slice_ne n γ : Proper (dist n ==> dist n) (slice N γ). Global Instance slice_ne n γ : Proper (dist n ==> dist n) (slice N γ). Proof. solve_proper. Qed. Proof. solve_proper. Qed. Global Instance box_ne n f : Proper (dist n ==> dist n) (box N f). Global Instance box_contractive f : Contractive (box N f). Proof. solve_proper. Qed. Proof. intros n P1 P2 HP1P2. apply exist_ne. intros Φ. f_equiv; last done. apply contractive. intros n' ?. by rewrite HP1P2. Qed. Global Instance slice_persistent γ P : PersistentP (slice N γ P). Global Instance slice_persistent γ P : PersistentP (slice N γ P). Proof. apply _. Qed. Proof. apply _. Qed. ... @@ -85,7 +89,7 @@ Proof. ... @@ -85,7 +89,7 @@ Proof. - by rewrite big_sepM_empty. - by rewrite big_sepM_empty. Qed. Qed. Lemma box_insert_empty E f P Q : Lemma box_insert_empty Q E f P : ▷ box N f P ={E}=∗ ∃ γ, ⌜f !! γ = None⌝ ∗ ▷ box N f P ={E}=∗ ∃ γ, ⌜f !! γ = None⌝ ∗ slice N γ Q ∗ ▷ box N (<[γ:=false]> f) (Q ∗ P). slice N γ Q ∗ ▷ box N (<[γ:=false]> f) (Q ∗ P). Proof. Proof. ... @@ -107,10 +111,10 @@ Qed. ... @@ -107,10 +111,10 @@ Qed. Lemma box_delete_empty E f P Q γ : Lemma box_delete_empty E f P Q γ : ↑N ⊆ E → ↑N ⊆ E → f !! γ = Some false → f !! γ = Some false → slice N γ Q ∗ ▷ box N f P ={E}=∗ ∃ P', slice N γ Q -∗ ▷ box N f P ={E}=∗ ∃ P', ▷ ▷ (P ≡ (Q ∗ P')) ∗ ▷ box N (delete γ f) P'. ▷ ▷ (P ≡ (Q ∗ P')) ∗ ▷ box N (delete γ f) P'. Proof. Proof. iIntros (??) "[#Hinv H]"; iDestruct "H" as (Φ) "[#HeqP Hf]". iIntros (??) "#Hinv H"; iDestruct "H" as (Φ) "[#HeqP Hf]". iExists ([∗ map] γ'↦_ ∈ delete γ f, Φ γ')%I. iExists ([∗ map] γ'↦_ ∈ delete γ f, Φ γ')%I. iInv N as (b) "(Hγ & #HγQ &_)" "Hclose". iInv N as (b) "(Hγ & #HγQ &_)" "Hclose". iApply fupd_trans_frame; iFrame "Hclose"; iModIntro; iNext. iApply fupd_trans_frame; iFrame "Hclose"; iModIntro; iNext. ... @@ -127,9 +131,9 @@ Qed. ... @@ -127,9 +131,9 @@ Qed. Lemma box_fill E f γ P Q : Lemma box_fill E f γ P Q : ↑N ⊆ E → ↑N ⊆ E → f !! γ = Some false → f !! γ = Some false → slice N γ Q ∗ ▷ Q ∗ ▷ box N f P ={E}=∗ ▷ box N (<[γ:=true]> f) P. slice N γ Q -∗ ▷ Q -∗ ▷ box N f P ={E}=∗ ▷ box N (<[γ:=true]> f) P. Proof. Proof. iIntros (??) "(#Hinv & HQ & H)"; iDestruct "H" as (Φ) "[#HeqP Hf]". iIntros (??) "#Hinv HQ H"; iDestruct "H" as (Φ) "[#HeqP Hf]". iInv N as (b') "(>Hγ & #HγQ & _)" "Hclose". iInv N as (b') "(>Hγ & #HγQ & _)" "Hclose". iDestruct (big_sepM_later _ f with "Hf") as "Hf". iDestruct (big_sepM_later _ f with "Hf") as "Hf". iDestruct (big_sepM_delete _ f _ false with "Hf") iDestruct (big_sepM_delete _ f _ false with "Hf") ... @@ -146,9 +150,9 @@ Qed. ... @@ -146,9 +150,9 @@ Qed. Lemma box_empty E f P Q γ : Lemma box_empty E f P Q γ : ↑N ⊆ E → ↑N ⊆ E → f !! γ = Some true → f !! γ = Some true → slice N γ Q ∗ ▷ box N f P ={E}=∗ ▷ Q ∗ ▷ box N (<[γ:=false]> f) P. slice N γ Q -∗ ▷ box N f P ={E}=∗ ▷ Q ∗ ▷ box N (<[γ:=false]> f) P. Proof. Proof. iIntros (??) "[#Hinv H]"; iDestruct "H" as (Φ) "[#HeqP Hf]". iIntros (??) "#Hinv H"; iDestruct "H" as (Φ) "[#HeqP Hf]". iInv N as (b) "(>Hγ & #HγQ & HQ)" "Hclose". iInv N as (b) "(>Hγ & #HγQ & HQ)" "Hclose". iDestruct (big_sepM_later _ f with "Hf") as "Hf". iDestruct (big_sepM_later _ f with "Hf") as "Hf". iDestruct (big_sepM_delete _ f with "Hf") iDestruct (big_sepM_delete _ f with "Hf") ... @@ -163,36 +167,35 @@ Proof. ... @@ -163,36 +167,35 @@ Proof. iFrame; eauto. iFrame; eauto. Qed. Qed. Lemma box_insert_full E f P Q : Lemma box_insert_full Q E f P : ↑N ⊆ E → ↑N ⊆ E → ▷ Q ∗ ▷ box N f P ={E}=∗ ∃ γ, ⌜f !! γ = None⌝ ∗ ▷ Q -∗ ▷ box N f P ={E}=∗ ∃ γ, ⌜f !! γ = None⌝ ∗ slice N γ Q ∗ ▷ box N (<[γ:=true]> f) (Q ∗ P). slice N γ Q ∗ ▷ box N (<[γ:=true]> f) (Q ∗ P). Proof. Proof. iIntros (?) "[HQ Hbox]". iIntros (?) "HQ Hbox". iMod (box_insert_empty with "Hbox") as (γ) "(% & #Hslice & Hbox)". iMod (box_insert_empty with "Hbox") as (γ) "(% & #Hslice & Hbox)". iExists γ. iFrame "%#". iExists γ. iFrame "%#". iMod (box_fill with "Hslice HQ Hbox"). done. iMod (box_fill with "[\$Hslice \$HQ \$Hbox]"). done. by apply lookup_insert. by rewrite insert_insert. by apply lookup_insert. by rewrite insert_insert. Qed. Qed. Lemma box_delete_full E f P Q γ : Lemma box_delete_full E f P Q γ : ↑N ⊆ E → ↑N ⊆ E → f !! γ = Some true → f !! γ = Some true → slice N γ Q ∗ ▷ box N f P ={E}=∗ slice N γ Q -∗ ▷ box N f P ={E}=∗ ▷ Q ∗ ∃ P', ▷ ▷ (P ≡ (Q ∗ P')) ∗ ▷ box N (delete γ f) P'. ∃ P', ▷ Q ∗ ▷ ▷ (P ≡ (Q ∗ P')) ∗ ▷ box N (delete γ f) P'. Proof. Proof. iIntros (??) "[#Hslice Hbox]". iIntros (??) "#Hslice Hbox". iMod (box_empty with "[\$Hslice \$Hbox]") as "[\$ Hbox]"; try done. iMod (box_empty with "Hslice Hbox") as "[\$ Hbox]"; try done. iMod (box_delete_empty with "[\$Hslice \$Hbox]") as (P') "[Heq Hbox]". iMod (box_delete_empty with "Hslice Hbox") as (P') "[Heq Hbox]". done. by apply lookup_insert. done. by apply lookup_insert. iExists P'. iFrame. rewrite -insert_delete delete_insert ?lookup_delete //. iExists P'. iFrame. rewrite -insert_delete delete_insert ?lookup_delete //. Qed. Qed. Lemma box_fill_all E f P : Lemma box_fill_all E f P : ↑N ⊆ E → ↑N ⊆ E → box N f P ∗ ▷ P ={E}=∗ box N (const true <\$> f) P. box N f P -∗ ▷ P ={E}=∗ box N (const true <\$> f) P. Proof. Proof. iIntros (?) "[H HP]"; iDestruct "H" as (Φ) "[#HeqP Hf]". iIntros (?) "H HP"; iDestruct "H" as (Φ) "[#HeqP Hf]". iExists Φ; iSplitR; first by rewrite big_sepM_fmap. iExists Φ; iSplitR; first by rewrite big_sepM_fmap. rewrite internal_eq_iff later_iff big_sepM_later. rewrite internal_eq_iff later_iff big_sepM_later. iDestruct ("HeqP" with "HP") as "HP". iDestruct ("HeqP" with "HP") as "HP". ... @@ -226,6 +229,54 @@ Proof. ... @@ -226,6 +229,54 @@ Proof. - rewrite internal_eq_iff later_iff big_sepM_later. by iApply "HeqP". - rewrite internal_eq_iff later_iff big_sepM_later. by iApply "HeqP". - iExists Φ; iSplit; by rewrite big_sepM_fmap. - iExists Φ; iSplit; by rewrite big_sepM_fmap. Qed. Qed. Lemma box_split E f P Q1 Q2 γ b : ↑N ⊆ E → f !! γ = Some b → slice N γ (Q1 ∗ Q2) -∗ ▷ box N f P ={E}=∗ ∃ γ1 γ2, ⌜delete γ f !! γ1 = None⌝ ∗ ⌜delete γ f !! γ2 = None⌝ ∗ slice N γ1 Q1 ∗ slice N γ2 Q2 ∗ ▷ box N (<[γ2 := b]>(<[γ1 := b]>(delete γ f))) P. Proof. iIntros (??) "#Hslice Hbox". destruct b. - iMod (box_delete_full with "Hslice Hbox") as (P') "([HQ1 HQ2] & Heq & Hbox)"; try done. iMod (box_insert_full Q1 with "HQ1 Hbox") as (γ1) "(% & #Hslice1 & Hbox)". done. iMod (box_insert_full Q2 with "HQ2 Hbox") as (γ2) "(% & #Hslice2 & Hbox)". done. iExists γ1, γ2. iFrame "%#". iModIntro. iSplit. { iPureIntro. by eapply lookup_insert_None. } iNext. eapply internal_eq_rewrite_contractive; [by apply _| |by eauto]. iNext. iRewrite "Heq". iPureIntro. rewrite assoc. f_equiv. by rewrite comm. done. - iMod (box_delete_empty with "Hslice Hbox") as (P') "[Heq Hbox]"; try done. iMod (box_insert_empty Q1 with "Hbox") as (γ1) "(% & #Hslice1 & Hbox)". iMod (box_insert_empty Q2 with "Hbox") as (γ2) "(% & #Hslice2 & Hbox)". iExists γ1, γ2. iFrame "%#". iModIntro. iSplit. { iPureIntro. by eapply lookup_insert_None. } iNext. eapply internal_eq_rewrite_contractive; [by apply _| |by eauto]. iNext. iRewrite "Heq". iPureIntro. rewrite assoc. f_equiv. by rewrite comm. done. Qed. Lemma box_combine E f P Q1 Q2 γ1 γ2 b : ↑N ⊆ E → γ1 ≠ γ2 → f !! γ1 = Some b → f !! γ2 = Some b → slice N γ1 Q1 -∗ slice N γ2 Q2 -∗ ▷ box N f P ={E}=∗ ∃ γ, ⌜delete γ2 (delete γ1 f) !! γ = None⌝ ∗ slice N γ (Q1 ∗ Q2) ∗ ▷ box N (<[γ := b]>(delete γ2 (delete γ1 f))) P. Proof. iIntros (????) "#Hslice1 #Hslice2 Hbox". destruct b. - iMod (box_delete_full with "Hslice1 Hbox") as (P1) "(HQ1 & Heq1 & Hbox)"; try done. iMod (box_delete_full with "Hslice2 Hbox") as (P2) "(HQ2 & Heq2 & Hbox)". done. by simplify_map_eq. iMod (box_insert_full (Q1 ∗ Q2)%I with "[\$HQ1 \$HQ2] Hbox") as (γ) "(% & #Hslice & Hbox)". done. iExists γ. iFrame "%#". iModIntro. iNext. eapply internal_eq_rewrite_contractive; [by apply _| |by eauto]. iNext. iRewrite "Heq1". iRewrite "Heq2". by rewrite assoc. - iMod (box_delete_empty with "Hslice1 Hbox") as (P1) "(Heq1 & Hbox)"; try done. iMod (box_delete_empty with "Hslice2 Hbox") as (P2) "(Heq2 & Hbox)". done. by simplify_map_eq. iMod (box_insert_empty (Q1 ∗ Q2)%I with "Hbox") as (γ) "(% & #Hslice & Hbox)". iExists γ. iFrame "%#". iModIntro. iNext. eapply internal_eq_rewrite_contractive; [by apply _| |by eauto]. iNext. iRewrite "Heq1". iRewrite "Heq2". by rewrite assoc. Qed. End box. End box. Typeclasses Opaque slice box. Typeclasses Opaque slice box.
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