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Proof step 7 from proof afs-cons-remove-orphans (previous step)

files[ino] = mkfile(m, md, m0, content), afs-cons(dirs, files) ⊦ (ino ∈ dirs ∧ ¬ ino ∈ orphans(dirs, files) ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dirs[ino].entries = ∅) ∧ ( ino ∈ dirs ∧ ¬ ino ∈ orphans(dirs, files) → dirs[ino].nlink = # links(ino, dirs) ∧ dirs[ino].nlink ≤ 1 ∧ dirs[ino].size = # dirs[ino].entries ∧ dirs[ino].nsubdirs = # subdirs(ino, dirs)) ∧ (ino ∈ files ∧ ¬ ino ∈ orphans(dirs, files) → ¬ ino ∈ dirs ∧ m0 = # links(ino, dirs)) ∧ ( ino ∈ dirs ∧ ¬ ino ∈ orphans(dirs, files) ∧ name ∈ dirs[ino].entries → dirs[ino].entries[name] ∈ dirs ∧ ¬ dirs[ino].entries[name] ∈ orphans(dirs, files) ∨ dirs[ino].entries[name] ∈ files ∧ ¬ dirs[ino].entries[name] ∈ orphans(dirs, files))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 8:
files[ino] = mkfile(m, md, m0, content), dirs[ino] = mkdir(m1, md0, m2, m3, entries), afs-cons(dirs, files) ⊦ (ino ∈ dirs ∧ ¬ ino ∈ orphans(dirs, files) ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → entries = ∅) ∧ (ino ∈ dirs ∧ ¬ ino ∈ orphans(dirs, files) → m2 = # links(ino, dirs) ∧ m2 ≤ 1 ∧ m1 = # entries ∧ m3 = # subdirs(ino, dirs)) ∧ (ino ∈ files ∧ ¬ ino ∈ orphans(dirs, files) → ¬ ino ∈ dirs ∧ m0 = # links(ino, dirs)) ∧ ( ino ∈ dirs ∧ ¬ ino ∈ orphans(dirs, files) ∧ name ∈ entries → entries[name] ∈ dirs ∧ ¬ entries[name] ∈ orphans(dirs, files) ∨ entries[name] ∈ files ∧ ¬ entries[name] ∈ orphans(dirs, files))

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[ino], m1, md0, m2, m3, entries]

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[ino], m1, md0, m2, m3, entries