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Proof step 133 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), files[entries2[str0]] = mkfile(m14, md8, m16, content), dirs0[entries2[str0]] = mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1), dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries2[str] ∈ dirs0, str0 ∈ entries2, str ∈ entries2, m ∈ dirs0, entries2[str0] ∈ dirs0, mklink(m, str0) ∈ links(entries2[str0], dirs0), # links(entries2[str0], dirs0) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries2[str] = ROOT_INO, entries2[str] = m, m21 = #(links(entries2[str0], dirs0) -- mklink(m, str0)) + 1, mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0]), mklink(m, str) ≠ mklink(m, str0) ∧ entries2[str0] = entries2[str]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 134:
# 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), files[entries2[str0]] = mkfile(m14, md8, m16, content), dirs0[entries2[str0]] = mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1), dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries2[str] ∈ dirs0, str ∈ entries2, m ∈ dirs0, entries2[str0] ∈ dirs0, str0 ∈ entries2, # links(entries2[str0], dirs0) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ # links(entries2[str0], dirs0) = m21, entries2[str] = ROOT_INO, entries2[str] = m, 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)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)%n ≤ m ⊦ (m - n) + n0 = n1 ↔ m + n0 = n + n1; (s)%⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n, n0, entries) ∧ str ∈ entries → links(entries[str], dirs) ≠ ∅; (s)%size-zero-02: ⊦ # 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)% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)% ⊦ alink ∈ ls → #(ls -- alink) = # ls - 1; (s)