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Proof step 140 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries2[str], dirs0) = m12 + 1, (ino = entries2[str] ⊃ mkdir(m13, md, m12, m15, ∅);mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4)) = mkdir(m20, md10, m21, m22, entries5), dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), files[ino] = mkfile(m14, md8, m16, content), dirs0[ino] = mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4), ino = entries2[str], afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries2[str] ∈ dirs0, str0 ∈ entries2, entries2[str0] ∈ dirs0, str ∈ entries2, m ∈ dirs0, ino ∈ dirs0, # links(ino, dirs0) ≤ 1, name ∈ entries4 → entries4[name] ∈ dirs0 ∨ entries4[name] ∈ files, ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries4 = ∅, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries2[str0] = ino, entries2[str] = ROOT_INO, entries2[str] = m, ino = m, m21 = # (entries2[str] = ino ⊃ links(entries2[str], dirs0) -- mklink(m, str);links(ino, dirs0)), mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 141:
# links(entries2[str], dirs0) = 1, dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str]] = mkdir(0, md6, 1, # subdirs(entries2[str], dirs0), ∅), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), files[entries2[str]] = mkfile(m14, md8, m16, content), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, str0 ∈ entries2, entries2[str0] ∈ dirs0, str ∈ entries2, m ∈ dirs0, entries2[str] ∈ dirs0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ 0 = #(links(entries2[str], dirs0) -- mklink(m, str)), entries2[str] = ROOT_INO, entries2[str] = m, links(entries2[str], dirs0) = ∅ ++ mklink(m, str), mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0])

Rule argument

Additional information

Used simplifier rules: lastlink-rename: ⊦ # links(entries[str0], dirs) = 1 ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(m, md, n, n0, entries) ∧ str0 ∈ entries ∧ str1 ∈ entries → (entries[str0] = entries[str1] ↔ str0 = str1);% ⊦ ls -- alink = ∅ ↔ ls = ∅ ∨ ls = ∅ ++ alink; (ws)%size-zero: ⊦ # ls ≠ 0 → ls ≠ ∅; (forward) from specification set-basic[alink-set]% ⊦ ¬ str ∈ ∅; (s)%afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ # ∅ = 0; (s)%⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n, n0, entries) ∧ str ∈ entries → links(entries[str], dirs) ≠ ∅; (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)