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Proof step 122 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries[str], dirs0) = m12 + 1, (entries2[str0] = ino ⊃ (links(entries2[str0], dirs0) -- mklink(m0, str0)) ++ mklink(m, str);links(ino, dirs0)) = ∅, (ino = m ⊃ mkdir(m8, md1, # links(m, dirs0), m11, entries[str, entries2[str0]]) ;(ino = m0 ⊃ mkdir(m1, md2, # links(m0, dirs0), m2, entries2 -- str0) ;mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4) ) ) = mkdir(m20, md10, m21, m22, entries5), dirs0[ino] = mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4), files[ino] = mkfile(m14, md8, m16, content), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), dirs0[entries[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[m] = mkdir(m8, md5, # links(m, dirs0), m11, entries), dirs0[m0] = mkdir(m1 + 1, md0, # links(m0, dirs0), m2 + 1, entries2), afs-cons(dirs0, files), m ∈ dirs0, ino ∈ dirs0, str ∈ entries, entries2[str0] ∈ dirs0, str0 ∈ entries2, m0 ∈ dirs0, entries[str] ∈ dirs0, ROOT_INO ∈ dirs0, # links(ino, dirs0) ≤ 1, links(ino, dirs0) = ∅ → entries4 = ∅, name ∈ entries4 → entries4[name] ∈ dirs0 ∨ entries4[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries[str] = m0, entries[str] = m, entries[str] = ROOT_INO, entries2[str0] = ino, entries5 = ∅, ino = entries[str], ino = ROOT_INO, m = m0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 123:
# links(entries[str], dirs0) = m12 + 1, # links(ino, dirs0) = 0, (ino = m ⊃ mkdir(m8, md1, # links(m, dirs0), m11, entries[str, entries2[str0]]) ;(ino = m0 ⊃ mkdir(m1, md2, # links(m0, dirs0), m2, entries2 -- str0);mkdir(0, md9, 0, # subdirs(ino, dirs0), ∅)) ) = mkdir(m20, md10, m21, m22, entries5), dirs0[ino] = mkdir(0, md9, 0, # subdirs(ino, dirs0), ∅), files[ino] = mkfile(m14, md8, m16, content), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), dirs0[entries[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[m] = mkdir(m8, md5, # links(m, dirs0), m11, entries), dirs0[m0] = mkdir(m1 + 1, md0, # links(m0, dirs0), m2 + 1, entries2), afs-cons(dirs0, files), m ∈ dirs0, ino ∈ dirs0, str ∈ entries, entries2[str0] ∈ dirs0, str0 ∈ entries2, m0 ∈ dirs0, entries[str] ∈ dirs0, ROOT_INO ∈ dirs0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries[str] = m0, entries[str] = m, entries[str] = ROOT_INO, entries2[str0] = ino, entries5 = ∅, ino = entries[str], ino = ROOT_INO, m = m0

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ str ∈ ∅; (s)% ⊦ # ∅ = 0; (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%size-zero-01: ⊦ ls ≠ ∅ → # ls ≠ 0; (forward) from specification set-basic[alink-set]%size-zero-02: ⊦ # ls = 0 → ls = ∅; (forward) from specification set-basic[alink-set]% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)% ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)%⊦ afs-cons(dirs, files) → ( links(ino, dirs) = ∅ ↔ (ino ∈ dirs → dirs[ino].nlink = 0) ∧ (ino ∈ files → files[ino].nlink = 0)); (ws)