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Proof step 1 from proof afs-cons-init (conclusion)

⊦ afs-cons((∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)], ∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 2:
⊦ afs-cons(dirs, files) ↔ (∀ ino. ROOT_INO ∈ dirs ∧ (ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dir-empty(ino, dirs, files)) ∧ ( 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 → valid-ino(dirs[ino].entries[name], dirs, files)) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → valid-pagesize(ino, n, files)))
Proof step 3:
⊦ ∀ ino. ROOT_INO ∈ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)] ∧ ( ino ∈ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)] ∧ ino ≠ ROOT_INO ∧ links(ino, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)]) = ∅ → dir-empty(ino, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)], ∅)) ∧ ( ino ∈ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)] → ¬ ino ∈ ∅ ∧ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].nlink = # links(ino, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)]) ∧ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].nlink ≤ 1 ∧ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].size = # (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].entries ∧ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].nsubdirs = # subdirs(ino, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)])) ∧ ( ino ∈ ∅ → ¬ ino ∈ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)] ∧ (∅)[ino].nlink = # links(ino, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)])) ∧ (∀ name. ino ∈ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)] ∧ name ∈ (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].entries → valid-ino((∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)][ino].entries[name], (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)], ∅)) ∧ (∀ n. ino ∈ ∅ ∧ n ∈ (∅)[ino].content → valid-pagesize(ino, n, ∅))

Rule argument

spec: <toplevel> name: afs-cons seq: ⊦ afs-cons(dirs, files) ↔ (∀ ino. ROOT_INO ∈ dirs ∧ ( ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dir-empty(ino, dirs, files)) ∧ ( 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 → valid-ino(dirs[ino].entries[name], dirs, files)) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → valid-pagesize(ino, n, files))) subst: [files, dirs] → [∅, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)]] flags: <all paths>

Additional information

Used terms: ∅, (∅ ++ ROOT_INO)[ROOT_INO, mkdir(0, md, 0, 0, ∅)]