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Proof step 156 from proof refine-unlink (previous step)

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

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 157:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 158:
# 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, content), 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, content)]), 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, content)]), 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, content)]), 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, content)]), 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_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, content)], 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 ⊦ ∀ ino. ( ino ∈ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str], mkfile(m6, md0, m0, content0)] ↔ ino ∈ (cache_files[entries0[str], mkfile(m6, md0, m0, content)] ⊕ cache_pcache) ⊕ 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)) ]) ∧ ( ino ∈ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str], mkfile(m6, md0, m0, content0)] → ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str], mkfile(m6, md0, m0, content0)][ino] = ( (cache_files[entries0[str], mkfile(m6, md0, m0, content)] ⊕ cache_pcache) ⊕ 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)) ] )[ino ])

Rule argument

spec: heap, inst: file-heap name: ext seq: ⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino])) subst: [files, files0] → [((cache_files ⊕ cache_pcache) ⊕ cache_icache )[entries0[str], mkfile(m6, md0, m0, content0) ], ( cache_files [entries0[str], mkfile(m6, md0, m0, content) ] ⊕ cache_pcache) ⊕ 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)) ]] flags: <all paths>

Additional information

Used terms: ((cache_files ⊕ cache_pcache) ⊕ cache_icache )[entries0[str], mkfile(m6, md0, m0, content0) ], ( cache_files[entries0[str], mkfile(m6, md0, m0, content)] ⊕ cache_pcache) ⊕ 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)) ]