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Proof step 13 from proof afs-cons-reduce (previous step)

dirs1[ino] = mkdir(m6, md2, m7, m8, entries0), dirs0[ino] = mkdir(m3, md1, m4, m5, entries), files1[ino] = mkfile(m1, md0, m2, content0), files0[ino] = mkfile(m, md, m0, content), afs-cons(dirs1, files1), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files0, ino ∈ dirs0 → m4 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m3 = # entries ∧ m5 = # subdirs(ino, dirs0) ⊦ (ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → entries0 = ∅) ∧ (ino ∈ dirs1 → m7 = # links(ino, dirs1) ∧ # links(ino, dirs1) ≤ 1 ∧ m6 = # entries0 ∧ m8 = # subdirs(ino, dirs1)) ∧ (ino ∈ files1 → ¬ ino ∈ dirs1 ∧ m2 = # links(ino, dirs1)) ∧ (ino ∈ dirs1 ∧ name ∈ entries0 → entries0[name] ∈ dirs1 ∨ entries0[name] ∈ files1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ 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 15:
files0[ino] = mkfile(m, md, m0, content), files1[ino] = mkfile(m1, md0, m2, content0), dirs0[ino] = mkdir(m3, md1, m4, m5, entries), dirs1[ino] = mkdir(m6, md2, m7, m8, entries0), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, ino ∈ dirs0 → m4 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m3 = # entries ∧ m5 = # subdirs(ino, dirs0), ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files0, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅, ∀ ino. ROOT_INO ∈ dirs1 ∧ (ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → dir-empty(ino, dirs1, files1)) ∧ ( ino ∈ dirs1 → ¬ ino ∈ files1 ∧ dirs1[ino].nlink = # links(ino, dirs1) ∧ dirs1[ino].nlink ≤ 1 ∧ dirs1[ino].size = # dirs1[ino].entries ∧ dirs1[ino].nsubdirs = # subdirs(ino, dirs1)) ∧ (ino ∈ files1 → ¬ ino ∈ dirs1 ∧ files1[ino].nlink = # links(ino, dirs1)) ∧ (∀ name. ino ∈ dirs1 ∧ name ∈ dirs1[ino].entries → valid-ino(dirs1[ino].entries[name], dirs1, files1)) ∧ (∀ n. ino ∈ files1 ∧ n ∈ files1[ino].content → valid-pagesize(ino, n, files1)) ⊦ (ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → entries0 = ∅) ∧ (ino ∈ dirs1 → m7 = # links(ino, dirs1) ∧ # links(ino, dirs1) ≤ 1 ∧ m6 = # entries0 ∧ m8 = # subdirs(ino, dirs1)) ∧ (ino ∈ files1 → ¬ ino ∈ dirs1 ∧ m2 = # links(ino, dirs1)) ∧ (ino ∈ dirs1 ∧ name ∈ entries0 → entries0[name] ∈ dirs1 ∨ entries0[name] ∈ files1)

Rule argument

spec: afs-base 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] → [files1, dirs1] flags: <all paths>

Additional information

Used terms: files1, dirs1