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Proof step 11 from proof refine-link (previous step)

afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(cache_dirs[ino].entries[str], md0, false, ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[cache_dirs[ino].entries[str]].nlink, 0, m6), err = err0, m0 = cache_dirs[m].nlink, m1 = cache_dirs[m].nsubdirs, m2 = cache_dirs[m].size, m3 = cache_dirs[ino].entries[str], m4 = ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[cache_dirs[ino].entries[str]].nlink, new_dent = negdentry(str0), p_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), ino ∈ cache_dirs, str ∈ cache_dirs[ino].entries, m ∈ cache_dirs, cache_dirs[ino].entries[str] ∈ cache_files, ¬ str0 ∈ cache_dirs[m].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(cache_dirs[ino].entries[str], cache_dirs) ≠ ∅ ∨ cache_dirs[ino].entries[str] = ROOT_INO ⊦ ⟪cache_link#(mkdentry(str, m3); p_inode, c_inode, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨old_dent0 := negdentry(str0); p_inode0 := mkinode(m, md, true, m0, m1, m2); p_inode1 := mkinode(m3, md0, false, m4, 0, m6); afs_link#(mkdentry(str, m3); p_inode0, p_inode1, old_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ new_dent = old_dent0 ∧ err = err0)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 12:
((cache_files ⊕ cache_pcache) ⊕ cache_icache)[cache_dirs[ino].entries[str]] = mkfile(m5, md1, m7, content), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(cache_dirs[ino].entries[str], md0, false, m7, 0, m6), err = err0, m0 = cache_dirs[m].nlink, m1 = cache_dirs[m].nsubdirs, m2 = cache_dirs[m].size, m3 = cache_dirs[ino].entries[str], m4 = m7, new_dent = negdentry(str0), p_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), cache_dirs[ino].entries[str] ∈ cache_files, m ∈ cache_dirs, str ∈ cache_dirs[ino].entries, ino ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(cache_dirs[ino].entries[str], cache_dirs) ≠ ∅ ∨ cache_dirs[ino].entries[str] = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO ⊦ ⟪cache_link#(mkdentry(str, m3); p_inode, c_inode, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨old_dent0 := negdentry(str0); p_inode0 := mkinode(m, md, true, m0, m1, m2); p_inode1 := mkinode(m3, md0, false, m4, 0, m6); afs_link#(mkdentry(str, m3); p_inode0, p_inode1, old_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ new_dent = old_dent0 ∧ err = err0), str0 ∈ cache_dirs[m].entries

Rule argument

spec: abstract-file name: elim seq: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content) subst: [af, m, md, m0, content] → [((cache_files ⊕ cache_pcache) ⊕ cache_icache )[cache_dirs[ino].entries[str] ], m5, md1, m7, content]

Additional information

Used simplifier rules: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content); (elim) Used terms: ((cache_files ⊕ cache_pcache) ⊕ cache_icache )[cache_dirs[ino].entries[str] ], m5, md1, m7, content