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Proof step 8 from proof inv-recovery (previous step)

files1[ino] = mkfile(m, md, m0, content), afs-cons(dirs0, files1), ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅, ino ∈ dirs0 → dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0), ino ∈ files1 → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files1 ⊦ (ino ∈ dirs0 ∧ ¬ ino ∈ orphans(dirs0, files1) ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅) ∧ ( ino ∈ dirs0 ∧ ¬ ino ∈ orphans(dirs0, files1) → dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0)) ∧ (ino ∈ files1 ∧ ¬ ino ∈ orphans(dirs0, files1) → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0)) ∧ ( ino ∈ dirs0 ∧ ¬ ino ∈ orphans(dirs0, files1) ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∧ ¬ dirs0[ino].entries[name] ∈ orphans(dirs0, files1) ∨ dirs0[ino].entries[name] ∈ files1 ∧ ¬ dirs0[ino].entries[name] ∈ orphans(dirs0, files1))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
files1[ino] = mkfile(m, md, m0, content), dirs0[ino] = mkdir(m1, md0, m2, m3, entries), afs-cons(dirs0, files1), ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files1, ino ∈ files1 → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0), ino ∈ dirs0 → m2 = # links(ino, dirs0) ∧ m2 ≤ 1 ∧ m1 = # entries ∧ m3 = # subdirs(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅ ⊦ (ino ∈ dirs0 ∧ ¬ ino ∈ orphans(dirs0, files1) ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅) ∧ (ino ∈ dirs0 ∧ ¬ ino ∈ orphans(dirs0, files1) → m2 = # links(ino, dirs0) ∧ m2 ≤ 1 ∧ m1 = # entries ∧ m3 = # subdirs(ino, dirs0)) ∧ (ino ∈ files1 ∧ ¬ ino ∈ orphans(dirs0, files1) → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0)) ∧ ( ino ∈ dirs0 ∧ ¬ ino ∈ orphans(dirs0, files1) ∧ name ∈ entries → entries[name] ∈ dirs0 ∧ ¬ entries[name] ∈ orphans(dirs0, files1) ∨ entries[name] ∈ files1 ∧ ¬ entries[name] ∈ orphans(dirs0, files1))

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] → [dirs0[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: dirs0[ino], m1, md0, m2, m3, entries