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Proof step 192 from proof refine-unlink (previous step)

# links(entries0[str], cache_dirs) = m0 + 1, cache_dirs[m3] = mkdir(m9 + 1, md, # links(m3, cache_dirs), m4, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, m0 + 1, content0), cache_files[entries0[str]] = mkfile(m2, md3, m0 + 1, content1), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m4, entries0 -- str)], cache_files[entries0[str], mkfile(m6, md0, m0, content1)]), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache -- dentrykey(m3, str), cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m4, entries0 -- str)], cache_files[entries0[str], mkfile(m6, md0, m0, content1)]), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), m4, m9)) ], cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m4, entries0 -- str)], cache_files[entries0[str], mkfile(m6, md0, m0, content1)]), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m4, entries0 -- str)], cache_files[entries0[str], mkfile(m6, md0, m0, content1)]), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m4, entries0 -- str)], cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), m4, m9)) ], cache_pcache), entries0[str] ∈ cache_icache, entries0[str] ∈ cache_files, str ∈ entries0, m3 ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), cache_sync → synced (cache_files[entries0[str], mkfile(m6, md0, m0, content1)], cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), m4, m9)) ], cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ entries0[str] = m3, (n ∈ content0 ↔ n ∈ content1 ⊕ mkcontent(entries0[str], cache_pcache)) ∧ (n ∈ content0 → content0[n] = (content1 ⊕ mkcontent(entries0[str], cache_pcache))[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: cache-base name: at-icache-overwrite seq: ⊦ ino ∈ icache ∧ ¬ icache[ino].inode.directory → (files ⊕ icache)[ino] = mkfile (icache[ino].inode.size, icache[ino].inode.meta, icache[ino].inode.nlink, files[ino].content) subst: [ino, icache, files] → [entries0[str], cache_icache, cache_files ⊕ cache_pcache] flags: <all paths>

Additional information

Used simplifier rules: ⊦ (files ⊕ pcache)[ino] = mkfile (files[ino].size, files[ino].meta, files[ino].nlink, files[ino].content ⊕ mkcontent(ino, pcache)); (s)% ⊦ mkfile(m, md, m0, content).content = content; (s)%⊦ mkfile(m, md, m0, content) = mkfile(m1, md0, m2, content0) ↔ m = m1 ∧ md = md0 ∧ m0 = m2 ∧ content = content0; (ws)%⊦ ino ∈ icache ∧ icache[ino].inode.directory → (files ⊕ icache)[ino] = files[ino]; (s)% ⊦ mkfile(m, md, m0, content).nlink = m0; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: entries0[str], cache_icache, cache_files ⊕ cache_pcache