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Proof step 197 from proof afs-cons-rename-dir-dentry (previous step)

# subdirs(m0, dirs0) = m11 + 1, # entries2 = m20 + 1, # links(entries2[str], dirs0) = m12 + 1, dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), files[m0] = mkfile(m14, md8, m16, content), dirs0[m0] = mkdir(m20 + 1, md0, # links(m0, dirs0), m11 + 1, entries2), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries2[str] ∈ dirs0, str0 ∈ entries2, entries2[str0] ∈ dirs0, str ∈ entries2, m0 ∈ dirs0, # links(m0, dirs0) ≤ 1, name ∈ entries2 → entries2[name] ∈ dirs0 ∨ entries2[name] ∈ files, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries2[str] = ROOT_INO, entries2[str] = m0, entries2 = ∅, m20 = # (entries2 -- str0)[str, entries2[str0]], mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0]), m0 ≠ ROOT_INO ∧ links(m0, dirs0) = ∅

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 198:
⊦ ¬ str ∈ entries → # entries[str, ino] = # entries + 1

Rule argument

spec: store, inst: dir-entries name: Size-insert seq: ⊦ ¬ str ∈ entries → # entries[str, ino] = # entries + 1 subst: [str, entries, ino] → [str, entries2 -- str0, entries2[str0]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ str ∈ entries → #(entries -- str) = (# entries)-1; (s)% ⊦ n -1 = n - 1; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ¬ m < n → ¬ m + m0 < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n ≤ m ⊦ (m - n) + n0 = n1 ↔ m + n0 = n + n1; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ str ∈ entries -- str0 ↔ str ≠ str0 ∧ str ∈ entries; (s)% ⊦ str ∈ entries → # entries[str, ino] = # entries; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s) Used terms: str, entries2 -- str0, entries2[str0]