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Proof step 32 from proof afs-cons-mkdir-ino (previous step)

(dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)[ino0] = mkdir(m7, md3, m8, m9, entries1), dirs[ino0] = mkdir(m4, md2, m5, m6, entries0), files[ino0] = mkfile(m1, md1, m3, content), dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, ino0 ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, dirs), ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries0 = ∅, ino0 ∈ dirs → m5 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m4 = # entries0 ∧ m6 = # subdirs(ino0, dirs), links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino0 = ino, str ∈ entries, ino ∈ files, ino ∈ dirs, (ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries1 = ∅) ∧ ( ino0 ∈ dirs → m8 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m7 = # entries1 ∧ m9 = # subdirs(ino0, dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)) ∧ (ino0 ∈ dirs ∧ name ∈ entries1 → entries1[name] = ino ∨ entries1[name] ∈ dirs ∨ entries1[name] ∈ files)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
ino ∈ dirs, ino0 ≠ 0, ¬ ino0 ∈ dirs ⊦ ino ≠ ino0 → (dirs ++ ino0)[ino] = dirs[ino]
Proof step 34:
dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), files[ino0] = mkfile(m1, md1, m3, content), dirs[ino0] = mkdir(m4, md2, m5, m6, entries0), dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])][ino0] = mkdir(m7, md3, m8, m9, entries1), afs-cons(dirs, files), m ∈ dirs, ROOT_INO ∈ dirs, ino0 ∈ dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])], ino0 ∈ dirs → m5 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m4 = # entries0 ∧ m6 = # subdirs(ino0, dirs), ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries0 = ∅, ino0 ∈ files → ¬ ino0 ∈ dirs ∧ m3 = # links(ino0, dirs), ino0 ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino0 = ino, ino ∈ dirs, ino ∈ files, str ∈ entries, ino ∈ dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])], (ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries1 = ∅) ∧ ( ino0 ∈ dirs → m8 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m7 = # entries1 ∧ m9 = # subdirs(ino0, dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)) ∧ (ino0 ∈ dirs ∧ name ∈ entries1 → entries1[name] = ino ∨ entries1[name] ∈ dirs ∨ entries1[name] ∈ files)

Rule argument

spec: heap, inst: dir-heap name: at-alloc seq: ino ∈ dirs, ino0 ≠ 0, ¬ ino0 ∈ dirs ⊦ ino ≠ ino0 → (dirs ++ ino0)[ino] = dirs[ino] subst: [ino, ino0, dirs] → [ino0, ino, dirs [m, mkdir (m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino]) ]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s) Used terms: ino0, ino, dirs [m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino]) ]