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Proof step 211 from proof afs-cons-rename-dir-negdentry (previous step)

dirs0[m] = mkdir(# entries, md2, # links(m, dirs0), 0, entries), dirs0[m0] = mkdir(m2, md3, # links(m0, dirs0), m5, entries0), dirs0[entries[str]] = mkdir(m1, md, # links(entries[str], dirs0), m3, entries2), subdirs(m, dirs0) = ∅, ROOT_INO ∈ dirs0, entries[str] ∈ dirs0, str ∈ entries, m ∈ dirs0, m0 ∈ dirs0, # links(m, dirs0) ≤ # entries, # links(m, dirs0) ≤ 1, 1 ≤ # entries, m0 = m → md1 = md0, ∀ ino. (ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅) ∧ ( ino ∈ dirs0 → ¬ ino ∈ files ∧ dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0)) ∧ (ino ∈ files → ¬ ino ∈ dirs0 ∧ files[ino].nlink = # links(ino, dirs0)) ∧ (∀ name. ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE), ∀ name. name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries = ∅, m ∈ files, str0 ∈ entries0, m ≠ ROOT_INO ∧ links(m, dirs0) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 212:
⊦ strings = strings0 ↔ (∀ str. str ∈ strings ↔ str ∈ strings0)
Proof step 213:
dirs0[entries[str]] = mkdir(m1, md, # links(entries[str], dirs0), m3, entries2), dirs0[m0] = mkdir(m2, md3, # links(m0, dirs0), m5, entries0), dirs0[m] = mkdir(# entries, md2, # links(m, dirs0), 0, entries), m0 ∈ dirs0, m ∈ dirs0, str ∈ entries, entries[str] ∈ dirs0, ROOT_INO ∈ dirs0, 1 ≤ # entries, # links(m, dirs0) ≤ 1, # links(m, dirs0) ≤ # entries, m0 = m → md1 = md0, ∀ name. name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files, ∀ ino. (ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅) ∧ ( ino ∈ dirs0 → ¬ ino ∈ files ∧ dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0)) ∧ (ino ∈ files → ¬ ino ∈ dirs0 ∧ files[ino].nlink = # links(ino, dirs0)) ∧ (∀ name. ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE), ∀ str. str ∈ subdirs(m, dirs0) ↔ str ∈ ∅, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries = ∅, str0 ∈ entries0, m ∈ files, m ≠ ROOT_INO ∧ links(m, dirs0) = ∅

Rule argument

spec: set-basic, inst: dir-entries name: Extension seq: ⊦ strings = strings0 ↔ (∀ str. str ∈ strings ↔ str ∈ strings0) subst: [strings0, strings] → [∅, subdirs(m, dirs0)] flags: <all paths>

Additional information

Used terms: ∅, subdirs(m, dirs0)