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Proof step 22 from proof afs-cons-mkdir-ino (previous step)

dirs[ino0] = mkdir(m4, md2, m5, m6, entries0), files[ino0] = mkfile(m1, md1, m3, content), dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, ino0 ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, dirs), ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries0 = ∅, ino0 ∈ dirs → m5 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m4 = # entries0 ∧ m6 = # subdirs(ino0, dirs), links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, str ∈ entries, ino ∈ dirs, ino ∈ files, ( (ino0 = ino ∨ ino0 ∈ dirs) ∧ ino0 ≠ ROOT_INO ∧ links(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)]) = ∅ → (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).entries = ∅) ∧ ( ino0 = ino ∨ ino0 ∈ dirs → ¬ ino0 ∈ files ∧ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).nlink = # links(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)]) ∧ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).nlink ≤ 1 ∧ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).size = # (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).entries ∧ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).nsubdirs = # subdirs(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)])) ∧ ( ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)])) ∧ ( (ino0 = ino ∨ ino0 ∈ dirs) ∧ name ∈ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).entries → (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).entries [name ] = ino ∨ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).entries [name ] ∈ dirs ∨ (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]).entries [name ] ∈ files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 23:
(ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅);(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0]) = mkdir(m7, md3, m8, m9, entries1), dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), files[ino0] = mkfile(m1, md1, m3, content), dirs[ino0] = mkdir(m4, md2, m5, m6, entries0), afs-cons(dirs, files), m ∈ dirs, ROOT_INO ∈ dirs, ino0 ∈ dirs → m5 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m4 = # entries0 ∧ m6 = # subdirs(ino0, dirs), ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries0 = ∅, ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, dirs), ino0 ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino ∈ files, ino ∈ dirs, str ∈ entries, ( (ino0 = ino ∨ ino0 ∈ dirs) ∧ ino0 ≠ ROOT_INO ∧ links(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)]) = ∅ → entries1 = ∅) ∧ ( ino0 = ino ∨ ino0 ∈ dirs → ¬ ino0 ∈ files ∧ m8 = # links(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)]) ∧ m8 ≤ 1 ∧ m7 = # entries1 ∧ m9 = # subdirs(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)])) ∧ ( ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, (dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino, mkdir(0, md, 1, 0, ∅)])) ∧ ((ino0 = ino ∨ ino0 ∈ dirs) ∧ name ∈ entries1 → entries1[name] = ino ∨ 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] → [(ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅) ;( dirs [m, mkdir (m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino]) ] ++ ino )[ino0 ] ), m7, md3, m8, m9, 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: (ino0 = ino ⊃ mkdir(0, md, 1, 0, ∅) ;( dirs [m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino]) ] ++ ino )[ino0 ] ), m7, md3, m8, m9, entries1