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Proof step 19 from proof afs-cons-create-ino (previous step)

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 ∈ dirs[ino0].entries → dirs[ino0].entries[name] ∈ dirs ∨ dirs[ino0].entries[name] ∈ files, ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, dirs), ino0 ∈ dirs → dirs[ino0].nlink = # links(ino0, dirs) ∧ dirs[ino0].nlink ≤ 1 ∧ dirs[ino0].size = # dirs[ino0].entries ∧ dirs[ino0].nsubdirs = # subdirs(ino0, dirs), ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → dirs[ino0].entries = ∅, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino ∈ files, ino ∈ dirs, str ∈ entries, ( ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries = ∅) ∧ ( ino0 ∈ dirs → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nlink = # links(ino0, dirs) ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nlink ≤ 1 ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).size = # (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nsubdirs = # subdirs(ino0, dirs[m, mkdir(m0, md0, # links(m, dirs), m2, entries[str, ino])])) ∧ ( ino0 = ino ∨ ino0 ∈ files → ¬ ino0 ∈ dirs ∧ (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]).nlink = # (ino = ino0 ⊃ ∅ ++ mklink(m, str);links(ino0, dirs))) ∧ ( ino0 ∈ dirs ∧ name ∈ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] ∈ dirs ∨ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] = ino ∨ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] ∈ files) ∧ ( (ino0 = ino ∨ ino0 ∈ files) ∧ n ∈ (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]).content → # (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]).content[n] = VFS_PAGE_SIZE)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 20:
(ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]) = mkfile(m4, md2, m5, content0), dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), files[ino0] = mkfile(m1, md1, m3, content), afs-cons(dirs, files), m ∈ dirs, ROOT_INO ∈ dirs, ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → dirs[ino0].entries = ∅, ino0 ∈ dirs → dirs[ino0].nlink = # links(ino0, dirs) ∧ dirs[ino0].nlink ≤ 1 ∧ dirs[ino0].size = # dirs[ino0].entries ∧ dirs[ino0].nsubdirs = # subdirs(ino0, dirs), ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, dirs), ino0 ∈ dirs ∧ name ∈ dirs[ino0].entries → dirs[ino0].entries[name] ∈ dirs ∨ dirs[ino0].entries[name] ∈ files, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, str ∈ entries, ino ∈ dirs, ino ∈ files, ( ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries = ∅) ∧ ( ino0 ∈ dirs → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nlink = # links(ino0, dirs) ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nlink ≤ 1 ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).size = # (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nsubdirs = # subdirs(ino0, dirs[m, mkdir(m0, md0, # links(m, dirs), m2, entries[str, ino])])) ∧ (ino0 = ino ∨ ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m5 = # (ino = ino0 ⊃ ∅ ++ mklink(m, str);links(ino0, dirs))) ∧ ( ino0 ∈ dirs ∧ name ∈ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] ∈ dirs ∨ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] = ino ∨ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] ∈ files) ∧ ((ino0 = ino ∨ ino0 ∈ files) ∧ n ∈ content0 → # content0[n] = VFS_PAGE_SIZE)

Rule argument

spec: abstract-file name: elim seq: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content) subst: [af, m, md, m0, content] → [(ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]), m3, md2, m4, content0]

Additional information

Used simplifier rules: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content); (elim) Used terms: (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]), m3, md2, m4, content0