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Proof step 44 from proof inv-link (previous step)

(ino = entries[str] ⊃ mkfile(m1, md4, # links(entries[str], dirs0) + 1, content0);mkfile(m3, md, m7, content)) = mkfile(m9, md0, m10, content1), dirs0[ino0] = mkdir(m2, md3, # links(ino0, dirs0), m6, entries0), files0[entries[str]] = mkfile(m1, md4, # links(entries[str], dirs0), content0), dirs0[m] = mkdir(m0, md1, # links(m, dirs0), m5, entries1), entries0[str] = entries[str], dirs0[ino1] = mkdir(m4, md2, # links(ino1, dirs0), m8, entries), files0[ino] = mkfile(m3, md, m7, content), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, entries[str] ∈ files0, m ∈ dirs0, str ∈ entries0, ino0 ∈ dirs0, ino1 ∈ dirs0, str ∈ entries, ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files0, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m7 = # links(ino, dirs0), 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 ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ str0 ∈ entries1, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0)) = ∅ → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nlink = # (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0)) ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nlink ≤ 1 ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).size = # (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nsubdirs = # subdirs(ino, dirs0[m, mkdir(m0, md1, # links(m, dirs0), m5, entries1[str0, entries[str]])])) ∧ (ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m10 = # (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0))) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries[name] ∈ dirs0 ∨ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries[name] ∈ files0) ∧ (ino ∈ files0 ∧ n ∈ content1 → # content1[n] = VFS_PAGE_SIZE)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 45:
(ino = entries[str] ⊃ mkfile(m1, md4, # links(entries[str], dirs0) + 1, content0);mkfile(m3, md, m7, content)) = mkfile(m9, md0, m10, content1), files0[ino] = mkfile(m3, md, m7, content), dirs0[ino1] = mkdir(m4, md2, # links(ino1, dirs0), m8, entries), entries0[str] = entries[str], dirs0[m] = mkdir(m0, md1, # links(m, dirs0), m5, entries1), files0[entries[str]] = mkfile(m1, md4, # links(entries[str], dirs0), content0), dirs0[ino0] = mkdir(m2, md3, # links(ino0, dirs0), m6, entries0), dirs0[ino] = mkdir(m11, md5, m12, m13, entries2), afs-cons(dirs0, files0), str ∈ entries, ino1 ∈ dirs0, ino0 ∈ dirs0, str ∈ entries0, m ∈ dirs0, entries[str] ∈ files0, ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries2 = ∅, ino ∈ dirs0 → m12 = # links(ino, dirs0) ∧ m12 ≤ 1 ∧ m11 = # entries2 ∧ m13 = # subdirs(ino, dirs0), ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m7 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ entries2 → entries2[name] ∈ dirs0 ∨ entries2[name] ∈ files0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ str0 ∈ entries1, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0)) = ∅ → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nlink = # (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0)) ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nlink ≤ 1 ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).size = # (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nsubdirs = # subdirs(ino, dirs0[m, mkdir(m0, md1, # links(m, dirs0), m5, entries1[str0, entries[str]])])) ∧ (ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m10 = # (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0))) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries[name] ∈ dirs0 ∨ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries[name] ∈ files0) ∧ (ino ∈ files0 ∧ n ∈ content1 → # content1[n] = VFS_PAGE_SIZE)

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], m11, md5, m12, m13, entries2]

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], m11, md5, m12, m13, entries2