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Proof step 20 from proof inv-write_inode (previous step)

dirs[ino] = mkdir(m2, md3, m6, m8, entries0), files[ino] = mkfile(m0, md2, m1, content0), files[m] = mkfile(m3, md0, m4, content), dirs[m] = mkdir(m5, md1, # links(m, dirs), m7, entries), afs-cons(dirs, files), m ∈ dirs, ROOT_INO ∈ dirs, ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → entries0 = ∅, ino ∈ files → ¬ ino ∈ dirs ∧ m1 = # links(ino, dirs), ino ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, ino ∈ dirs → m6 = # links(ino, dirs) ∧ # links(ino, dirs) ≤ 1 ∧ m2 = # entries0 ∧ m8 = # subdirs(ino, dirs) ⊦ ( ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).entries = ∅) ∧ ( ino ∈ dirs → (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).nlink = # links(ino, dirs) ∧ (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).nlink ≤ 1 ∧ (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).size = # (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).entries ∧ (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).nsubdirs = # subdirs(ino, dirs)) ∧ ( ino ∈ dirs ∧ name ∈ (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).entries → (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).entries[name] ∈ dirs ∨ (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)).entries[name] ∈ files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 21:
(ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries);mkdir(m2, md3, m6, m8, entries0)) = mkdir(m9, md4, m10, m11, entries1), dirs[m] = mkdir(m5, md1, # links(m, dirs), m7, entries), files[m] = mkfile(m3, md0, m4, content), files[ino] = mkfile(m0, md2, m1, content0), dirs[ino] = mkdir(m2, md3, m6, m8, entries0), afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, ino ∈ dirs → m6 = # links(ino, dirs) ∧ # links(ino, dirs) ≤ 1 ∧ m2 = # entries0 ∧ m8 = # subdirs(ino, dirs), ino ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, ino ∈ files → ¬ ino ∈ dirs ∧ m1 = # links(ino, dirs), ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → entries0 = ∅ ⊦ (ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → entries1 = ∅) ∧ (ino ∈ dirs → m10 = # links(ino, dirs) ∧ m10 ≤ 1 ∧ m9 = # entries1 ∧ m11 = # subdirs(ino, dirs)) ∧ (ino ∈ dirs ∧ name ∈ entries1 → entries1[name] ∈ dirs ∨ entries1[name] ∈ 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] → [(ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries) ;mkdir(m2, md3, m6, m8, entries0) ), m9, md4, m10, m11, 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: (ino = m ⊃ mkdir(m5, md, # links(m, dirs), m7, entries) ;mkdir(m2, md3, m6, m8, entries0) ), m9, md4, m10, m11, entries1