Commit da3aa7e4 authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan

Prove slice_iff.

parent e0a45a07
......@@ -239,6 +239,24 @@ Proof.
- iExists Φ; iSplit; by rewrite big_sepM_fmap.
Qed.
Lemma slice_iff E q f P Q Q' γ b :
N E f !! γ = Some b
(Q Q') - slice N γ Q - ?q box N f P ={E}= γ' P',
delete γ f !! γ' = None ?q (P P')
slice N γ' Q' ?q box N (<[γ' := b]>(delete γ f)) P'.
Proof.
iIntros (??) "#HQQ' #Hs Hb". destruct b.
- iMod (slice_delete_full with "Hs Hb") as (P') "(HQ & Heq & Hb)"; try done.
iDestruct ("HQQ'" with "HQ") as "HQ'".
iMod (slice_insert_full with "HQ' Hb") as (γ') "(% & #Hs' & Hb)"; try done.
iExists γ', _. iFrame "∗#%". iIntros "!>". do 2 iNext. iRewrite "Heq".
iAlways. by iSplit; iIntros "[? $]"; iApply "HQQ'".
- iMod (slice_delete_empty with "Hs Hb") as (P') "(Heq & Hb)"; try done.
iMod (slice_insert_empty with "Hb") as (γ') "(% & #Hs' & Hb)"; try done.
iExists γ', _. iFrame "∗#%". iIntros "!>". do 2 iNext. iRewrite "Heq".
iAlways. by iSplit; iIntros "[? $]"; iApply "HQQ'".
Qed.
Lemma slice_split E q f P Q1 Q2 γ b :
N E f !! γ = Some b
slice N γ (Q1 Q2) - ?q box N f P ={E}= γ1 γ2,
......
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