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Proof step 217 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries4[str], dirs0) = m12 + 1, files[m] = mkfile(m14, md8, m16, content), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), dirs0[entries4[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[m] = mkdir(# entries4, md9, # links(m, dirs0), # subdirs(m, dirs0), entries4), dirs0[m0] = mkdir(m1 + 1, md0, # links(m0, dirs0), m2 + 1, entries2), afs-cons(dirs0, files), m ∈ dirs0, str ∈ entries4, entries2[str0] ∈ dirs0, str0 ∈ entries2, m0 ∈ dirs0, entries4[str] ∈ dirs0, ROOT_INO ∈ dirs0, # links(m, dirs0) ≤ 1, name ∈ entries4 → entries4[name] ∈ dirs0 ∨ entries4[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ # subdirs(m, dirs0) = #(subdirs(m, dirs0) ++ str), entries4[str] = m0, entries4[str] = m, entries4[str] = ROOT_INO, entries4 = ∅, m0 = m, m ≠ ROOT_INO ∧ links(m, dirs0) = ∅

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-basic, inst: dir-entries name: Size-insert seq: ⊦ ¬ str ∈ strings → #(strings ++ str) = # strings + 1 subst: [str, strings] → [str, subdirs(m, dirs0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m = m + n ↔ n = 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%⊦ str ∈ subdirs(ino, dirs) ↔ ino ∈ dirs ∧ str ∈ dirs[ino].entries ∧ dirs[ino].entries[str] ∈ dirs; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ str ∈ strings → strings ++ str = strings; (s) Used terms: str, subdirs(m, dirs0)