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Proof step 123 from proof inv-rmdir (previous step)

dirs[m] = mkdir(# entries0, md0, # links(m, dirs), 0, entries0), subdirs(m, dirs) = ∅, ROOT_INO ∈ dirs, m ∈ dirs, str ∈ entries0, entries0[str] ∈ dirs, # links(m, dirs) ≤ 1, ∀ ino. (ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dirs[ino].entries = ∅) ∧ ( ino ∈ dirs → ¬ ino ∈ files ∧ dirs[ino].nlink = # links(ino, dirs) ∧ dirs[ino].nlink ≤ 1 ∧ dirs[ino].size = # dirs[ino].entries ∧ dirs[ino].nsubdirs = # subdirs(ino, dirs)) ∧ (ino ∈ files → ¬ ino ∈ dirs ∧ files[ino].nlink = # links(ino, dirs)) ∧ (∀ name. ino ∈ dirs ∧ name ∈ dirs[ino].entries → dirs[ino].entries[name] ∈ dirs ∨ dirs[ino].entries[name] ∈ files) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE), ∀ name. name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files ⊦ entries0 = ∅, m ∈ files, m ≠ ROOT_INO ∧ links(m, dirs) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 124:
⊦ strings = strings0 ↔ (∀ str. str ∈ strings ↔ str ∈ strings0)
Proof step 125:
dirs[m] = mkdir(# entries0, md0, # links(m, dirs), 0, entries0), entries0[str] ∈ dirs, str ∈ entries0, m ∈ dirs, ROOT_INO ∈ dirs, # links(m, dirs) ≤ 1, ∀ name. name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, ∀ ino. (ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dirs[ino].entries = ∅) ∧ ( ino ∈ dirs → ¬ ino ∈ files ∧ dirs[ino].nlink = # links(ino, dirs) ∧ dirs[ino].nlink ≤ 1 ∧ dirs[ino].size = # dirs[ino].entries ∧ dirs[ino].nsubdirs = # subdirs(ino, dirs)) ∧ (ino ∈ files → ¬ ino ∈ dirs ∧ files[ino].nlink = # links(ino, dirs)) ∧ (∀ name. ino ∈ dirs ∧ name ∈ dirs[ino].entries → dirs[ino].entries[name] ∈ dirs ∨ dirs[ino].entries[name] ∈ files) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE), ∀ str. str ∈ subdirs(m, dirs) ↔ str ∈ ∅ ⊦ entries0 = ∅, m ∈ files, m ≠ ROOT_INO ∧ links(m, dirs) = ∅

Rule argument

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

Additional information

Used terms: ∅, subdirs(m, dirs)