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Proof step 54 from proof inv-recover (previous step)

files1[ino] = mkfile(m, md, m0, content), dirs0[ino] = mkdir(# entries, md0, # links(ino, dirs0), # subdirs(ino, dirs0), entries), name ∈ entries, ino ∈ dirs0, ROOT_INO ∈ dirs0, # links(ino, dirs0) ≤ 1, ∀ ino0. (ino0 ∈ dirs0 ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs0) = ∅ → dirs0[ino0].entries = ∅) ∧ ( ino0 ∈ dirs0 → ¬ ino0 ∈ files1 ∧ dirs0[ino0].nlink = # links(ino0, dirs0) ∧ dirs0[ino0].nlink ≤ 1 ∧ dirs0[ino0].size = # dirs0[ino0].entries ∧ dirs0[ino0].nsubdirs = # subdirs(ino0, dirs0)) ∧ (ino0 ∈ files1 → ¬ ino0 ∈ dirs0 ∧ files1[ino0].nlink = # links(ino0, dirs0)) ∧ (∀ name0. ino0 ∈ dirs0 ∧ name0 ∈ dirs0[ino0].entries → dirs0[ino0].entries[name0] ∈ dirs0 ∨ dirs0[ino0].entries[name0] ∈ files1) ∧ (∀ n. ino0 ∈ files1 ∧ n ∈ files1[ino0].content → # files1[ino0].content[n] = VFS_PAGE_SIZE), ∀ name0. name0 ∈ entries → entries[name0] ∈ dirs0 ∨ entries[name0] ∈ files1 ⊦ entries = ∅, ino ∈ orphans(dirs0, files1), ino ∈ files1, ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅, entries[name] ∈ files1 ∧ ¬ entries[name] ∈ orphans(dirs0, files1), entries[name] ∈ dirs0 ∧ ¬ entries[name] ∈ orphans(dirs0, files1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ ino ∈ orphans(dirs, files) ↔ valid-ino(ino, dirs, files) ∧ links(ino, dirs) = ∅ ∧ ino ≠ ROOT_INO
Proof step 56:
dirs0[ino] = mkdir(# entries, md0, # links(ino, dirs0), # subdirs(ino, dirs0), entries), files1[ino] = mkfile(m, md, m0, content), ROOT_INO ∈ dirs0, ino ∈ dirs0, name ∈ entries, # links(ino, dirs0) ≤ 1, ∀ name0. name0 ∈ entries → entries[name0] ∈ dirs0 ∨ entries[name0] ∈ files1, ∀ ino0. (ino0 ∈ dirs0 ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs0) = ∅ → dirs0[ino0].entries = ∅) ∧ ( ino0 ∈ dirs0 → ¬ ino0 ∈ files1 ∧ dirs0[ino0].nlink = # links(ino0, dirs0) ∧ dirs0[ino0].nlink ≤ 1 ∧ dirs0[ino0].size = # dirs0[ino0].entries ∧ dirs0[ino0].nsubdirs = # subdirs(ino0, dirs0)) ∧ (ino0 ∈ files1 → ¬ ino0 ∈ dirs0 ∧ files1[ino0].nlink = # links(ino0, dirs0)) ∧ (∀ name0. ino0 ∈ dirs0 ∧ name0 ∈ dirs0[ino0].entries → dirs0[ino0].entries[name0] ∈ dirs0 ∨ dirs0[ino0].entries[name0] ∈ files1) ∧ (∀ n. ino0 ∈ files1 ∧ n ∈ files1[ino0].content → # files1[ino0].content[n] = VFS_PAGE_SIZE) ⊦ entries = ∅, ino ∈ files1, ino ∈ orphans(dirs0, files1), entries[name] ∈ dirs0 ∧ ¬(valid-ino(entries[name], dirs0, files1) ∧ links(entries[name], dirs0) = ∅ ∧ entries[name] ≠ ROOT_INO), entries[name] ∈ files1 ∧ ¬(valid-ino(entries[name], dirs0, files1) ∧ links(entries[name], dirs0) = ∅ ∧ entries[name] ≠ ROOT_INO), ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅

Rule argument

spec: afs-base name: orphans seq: ⊦ ino ∈ orphans(dirs, files) ↔ valid-ino(ino, dirs, files) ∧ links(ino, dirs) = ∅ ∧ ino ≠ ROOT_INO subst: [files, dirs, ino] → [files1, dirs0, entries[name]] flags: <all paths>

Additional information

Used terms: files1, dirs0, entries[name]