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Proof step 167 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries2[str], dirs0) = m12 + 1, (entries2[str0] = entries2[str] ⊃ mkdir(m13, md, m12, m15, ∅) ;(entries2[str0] = m ⊃ mkdir(m8, md1, # links(entries2[str0], dirs0), m11, (entries2 -- str0)[str, entries2[str0]]) ;mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1) ) ) = mkdir(m20, md10, m21, m22, entries5), dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str0]] = mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1), files[entries2[str0]] = mkfile(m14, md8, m16, content), afs-cons(dirs0, files), entries2[str0] ∈ dirs0, m ∈ dirs0, str ∈ entries2, str0 ∈ entries2, entries2[str] ∈ dirs0, ROOT_INO ∈ dirs0, # links(entries2[str0], dirs0) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ #((links(entries2[str0], dirs0) -- mklink(m, str0)) ++ mklink(m, str)) = 0, #((links(entries2[str0], dirs0) -- mklink(m, str0)) ++ mklink(m, str)) = 1, entries2[str] = m, entries2[str] = ROOT_INO, mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0]), mklink(m, str) ∈ links(entries2[str0], dirs0) -- mklink(m, str0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 168:
# links(entries2[str], dirs0) = m12 + 1, (entries2[str0] = entries2[str] ⊃ mkdir(m13, md, m12, m15, ∅) ;(entries2[str0] = m ⊃ mkdir(m8, md1, # links(entries2[str0], dirs0), m11, (entries2 -- str0)[str, entries2[str0]]) ;mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1) ) ) = mkdir(m20, md10, m21, m22, entries5), dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str0]] = mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1), files[entries2[str0]] = mkfile(m14, md8, m16, content), afs-cons(dirs0, files), entries2[str0] ∈ dirs0, m ∈ dirs0, str ∈ entries2, str0 ∈ entries2, entries2[str] ∈ dirs0, ROOT_INO ∈ dirs0, # links(entries2[str0], dirs0) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ #(links(entries2[str0], dirs0) -- mklink(m, str0)) = 0, entries2[str] = m, entries2[str] = ROOT_INO, links(entries2[str0], dirs0) = ∅ ++ mklink(m, str0), mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0]), mklink(m, str) ≠ mklink(m, str0) ∧ entries2[str0] = entries2[str]

Rule argument

Additional information

Used simplifier rules: ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)%⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n, n0, entries) ∧ str ∈ entries → links(entries[str], dirs) ≠ ∅; (s)% ⊦ ls -- alink = ∅ ↔ ls = ∅ ∨ ls = ∅ ++ alink; (ws)%size-zero: ⊦ # ls ≠ 0 → ls ≠ ∅; (forward) from specification set-basic[alink-set]% ⊦ mklink(m, str).name = str; (s)% ⊦ mklink(m, str).pino = m; (s)%⊦ alink ∈ links(ino, dirs) ↔ alink.pino ∈ dirs ∧ alink.name ∈ dirs[alink.pino].entries ∧ ino = dirs[alink.pino].entries[alink.name]; (s)% ⊦ alink ∈ ls -- alink0 ↔ alink ≠ alink0 ∧ alink ∈ ls; (s)% ⊦ m + 1 ≠ 0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws)% ⊦ ¬ alink ∈ ls → #(ls ++ alink) = # ls + 1; (s)