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Proof step 45 from proof fsync-pages-overwrite (previous step)

pcache[datakey(m, n)] = mkpentry(true, buf), files0[m] = mkfile(m1, md1, # links(m, dirs), content1), files[m] = mkfile(m1, md1, # links(m, dirs), content0), n * VFS_PAGE_SIZE < m2, afs-cons(dirs, files0), afs-cons(dirs, files), pcache-cons(pcache1, dirs, files0), pcache-cons(pcache, dirs, files), m ∈ files0, m ∈ files, datakey(m, n) ∈ pcache, ∀ key. (key ∈ pcache ↔ key ∈ pcache1) ∧ pcache[key].page = pcache1[key].page ∧ (pcache[key].dirty → pcache1[key].dirty), ∀ pageno. (pageno < n ⊃ (pageno ∈ content0 ↔ pageno ∈ content1 ⊕ mkcontent(m, pcache1)) ∧ content0[pageno] = (content1 ⊕ mkcontent(m, pcache1))[pageno] ;(pageno ∈ content0 ↔ pageno ∈ content1) ∧ content0[pageno] = content1[pageno] ) ⊦ # buf = VFS_PAGE_SIZE

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ pcache-cons(pcache, dirs, files) ↔ (∀ key. key ∈ pcache → key.data? ∧ # pcache[key].page = VFS_PAGE_SIZE ∧ valid-file-ino(key.ino, dirs, files) ∧ (¬ pcache[key].dirty → key.part ∈ files[key.ino].content ∧ pcache[key].page = files[key.ino].content[key.part]))
Proof step 47:
files[m] = mkfile(m1, md1, # links(m, dirs), content0), files0[m] = mkfile(m1, md1, # links(m, dirs), content1), pcache[datakey(m, n)] = mkpentry(true, buf), n * VFS_PAGE_SIZE < m2, afs-cons(dirs, files), afs-cons(dirs, files0), pcache-cons(pcache1, dirs, files0), datakey(m, n) ∈ pcache, m ∈ files, m ∈ files0, ∀ pageno. (pageno < n ⊃ (pageno ∈ content0 ↔ pageno ∈ content1 ⊕ mkcontent(m, pcache1)) ∧ content0[pageno] = (content1 ⊕ mkcontent(m, pcache1))[pageno] ;(pageno ∈ content0 ↔ pageno ∈ content1) ∧ content0[pageno] = content1[pageno] ), ∀ key. (key ∈ pcache ↔ key ∈ pcache1) ∧ pcache[key].page = pcache1[key].page ∧ (pcache[key].dirty → pcache1[key].dirty), ∀ key. key ∈ pcache → key.data? ∧ # pcache[key].page = VFS_PAGE_SIZE ∧ valid-file-ino(key.ino, dirs, files) ∧ (¬ pcache[key].dirty → key.part ∈ files[key.ino].content ∧ pcache[key].page = files[key.ino].content[key.part]) ⊦ # buf = VFS_PAGE_SIZE

Rule argument

spec: cache-base name: pcache-cons seq: ⊦ pcache-cons(pcache, dirs, files) ↔ (∀ key. key ∈ pcache → key.data? ∧ # pcache[key].page = VFS_PAGE_SIZE ∧ valid-file-ino(key.ino, dirs, files) ∧ ( ¬ pcache[key].dirty → key.part ∈ files[key.ino].content ∧ pcache[key].page = files[key.ino].content[key.part])) subst: [files, dirs, pcache] → [files, dirs, pcache] flags: <all paths>

Additional information

Used terms: files, dirs, pcache