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Proof step 87 from proof fsync-pages-overwrite (previous step)

# buf = VFS_PAGE_SIZE, (key0.ino = m ⊃ mkfile(m0, md, # links(key0.ino, dirs), content[n, buf]);files1[key0.ino]) = mkfile(m1, md0, m3, content0), pcache[datakey(m, n), mkpentry(false, buf)][key0] = mkpentry(boolvar1, buf2), pcache1[key] = mkpentry(boolvar0, buf1), pcache[datakey(m, n)] = mkpentry(true, buf), files0[m] = mkfile(m0, md, # links(m, dirs), content1), files1[m] = mkfile(m0, md, # links(m, dirs), content), pcache[datakey(m, n), mkpentry(false, buf)][key] = mkpentry(boolvar, buf0), n * VFS_PAGE_SIZE < m2, afs-cons(dirs, files1[m, mkfile(m0, md, # links(m, dirs), content[n, buf])]), afs-cons(dirs, files1), afs-cons(dirs, files0), pcache-cons(pcache1, dirs, files0), pcache-cons(pcache, dirs, files1), key0 ∈ pcache[datakey(m, n), mkpentry(false, buf)], m ∈ files0, m ∈ files1, 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 ∈ content ↔ pageno ∈ content1 ⊕ mkcontent(m, pcache1)) ∧ content[pageno] = (content1 ⊕ mkcontent(m, pcache1))[pageno] ;(pageno ∈ content ↔ pageno ∈ content1) ∧ content[pageno] = content1[pageno] ) ⊦ key0.data? ∧ # buf2 = VFS_PAGE_SIZE ∧ key0.ino ∈ files1 ∧ (¬ boolvar1 → key0.part ∈ content0 ∧ buf2 = content0[key0.part])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 88:
⊦ 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 89:
# buf = VFS_PAGE_SIZE, (key0.ino = m ⊃ mkfile(m0, md, # links(key0.ino, dirs), content[n, buf]);files1[key0.ino]) = mkfile(m1, md0, m3, content0), pcache[datakey(m, n), mkpentry(false, buf)][key] = mkpentry(boolvar, buf0), files1[m] = mkfile(m0, md, # links(m, dirs), content), files0[m] = mkfile(m0, md, # links(m, dirs), content1), pcache[datakey(m, n)] = mkpentry(true, buf), pcache1[key] = mkpentry(boolvar0, buf1), pcache[datakey(m, n), mkpentry(false, buf)][key0] = mkpentry(boolvar1, buf2), n * VFS_PAGE_SIZE < m2, afs-cons(dirs, files0), afs-cons(dirs, files1), afs-cons(dirs, files1[m, mkfile(m0, md, # links(m, dirs), content[n, buf])]), pcache-cons(pcache1, dirs, files0), datakey(m, n) ∈ pcache, m ∈ files1, m ∈ files0, key0 ∈ pcache[datakey(m, n), mkpentry(false, buf)], ∀ pageno. (pageno < n ⊃ (pageno ∈ content ↔ pageno ∈ content1 ⊕ mkcontent(m, pcache1)) ∧ content[pageno] = (content1 ⊕ mkcontent(m, pcache1))[pageno] ;(pageno ∈ content ↔ pageno ∈ content1) ∧ content[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, files1) ∧ (¬ pcache[key].dirty → key.part ∈ files1[key.ino].content ∧ pcache[key].page = files1[key.ino].content[key.part]) ⊦ key0.data? ∧ # buf2 = VFS_PAGE_SIZE ∧ key0.ino ∈ files1 ∧ (¬ boolvar1 → key0.part ∈ content0 ∧ buf2 = content0[key0.part])

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] → [files1, dirs, pcache] flags: <all paths>

Additional information

Used terms: files1, dirs, pcache