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Proof step 394 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries2[str0], dirs) = dirs[entries2[str0]].nlink, # links(m, dirs) = dirs[m].nlink, # links(entries[str], dirs) = dirs[entries[str]].nlink, dirs[entries[str]].entries = ∅, entries[str] = dirs[m].entries[str], dirs[m0] = mkdir(m1, md, # links(m0, dirs), m3, entries2), dirs[m] = mkdir(dirs[m].size, md4, dirs[m].nlink, dirs[m].nsubdirs, entries), dirs[entries[str]] = mkdir(dirs[entries[str]].size, md3, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, ∅), dirs[entries2[str0]] = mkdir(dirs[entries2[str0]].size, md5, dirs[entries2[str0]].nlink, dirs[entries2[str0]].nsubdirs, entries0), entries[str] ≠ ROOT_INO, afs-cons(dirs, files), str ∈ dirs[m].entries, m0 ∈ dirs, m ∈ dirs, entries2[str0] ∈ dirs, entries[str] ∈ dirs, str0 ∈ entries2, str ∈ entries, m = m0 → md1 = md2, links(m0, dirs) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪fail#(; ; err); asynced := ?⟫ true

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 395:
# links(entries[str], dirs) = dirs[entries[str]].nlink, # links(m, dirs) = m4, # links(entries2[str0], dirs) = dirs[entries2[str0]].nlink, dirs[entries[str]].entries = ∅, dirs[entries2[str0]] = mkdir(dirs[entries2[str0]].size, md5, dirs[entries2[str0]].nlink, dirs[entries2[str0]].nsubdirs, entries0), dirs[entries[str]] = mkdir(dirs[entries[str]].size, md3, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, ∅), dirs[m] = mkdir(m2, md4, m4, m5, entries), dirs[m0] = mkdir(m1, md, # links(m0, dirs), m3, entries2), entries[str] = entries1[str], dirs[m] = mkdir(m2, md0, m4, m5, entries1), afs-cons(dirs, files), str ∈ entries, str0 ∈ entries2, entries[str] ∈ dirs, entries2[str0] ∈ dirs, m ∈ dirs, m0 ∈ dirs, str ∈ entries1, m = m0 → md1 = md2, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ⟪fail#(; ; err); asynced := ?⟫ true, entries[str] = ROOT_INO

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [dirs[m], m2, md0, m4, m5, entries1]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: dirs[m], m2, md0, m4, m5, entries1