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Proof step 231 from proof fsync-pages-overwrite (previous step)

pageno ∈ content0 ↔ pageno ∈ content, content0[pageno] = content[pageno], files0[m] = mkfile(m0, md0, # links(m, dirs), content), files[m] = mkfile(m0, md0, # links(m, dirs), content0), pageno * 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, ∀ key. (key ∈ pcache ↔ key ∈ pcache1) ∧ pcache[key].page = pcache1[key].page ∧ (pcache[key].dirty → pcache1[key].dirty), ∀ pageno0. (pageno0 < pageno ⊃ (pageno0 ∈ content0 ↔ pageno0 ∈ content ⊕ mkcontent(m, pcache1)) ∧ content0[pageno0] = (content ⊕ mkcontent(m, pcache1))[pageno0] ;(pageno0 ∈ content0 ↔ pageno0 ∈ content) ∧ content0[pageno0] = content[pageno0] ) ⊦ content[pageno] = (content ⊕ mkcontent(m, pcache1))[pageno], datakey(m, pageno) ∈ pcache, datakey(m, pageno) ∈ pcache1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: cache-base name: at-overwrite-content-old seq: ⊦ ¬ n ∈ content1 → (content0 ⊕ content1)[n] = content0[n] subst: [n, content1, content0] → [pageno, mkcontent(m, pcache1), content] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ∈ content1 → (content0 ⊕ content1)[n] = content1[n]; (s)% ⊦ mkcontent(ino, pcache)[n] = pcache[datakey(ino, n)].page; (s)% ⊦ ¬ key ∈ pcache → pcache[key] = (∅)[key]; (s)% ⊦ n ∈ mkcontent(ino, pcache) ↔ datakey(ino, n) ∈ pcache; (s) Used terms: pageno, mkcontent(m, pcache1), content