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Proof step 130 from proof afs-cons-rename-file-dentry (previous step)

# entries1 = m19 + 1, # links(entries1[str], dirs0) = m11 + 1, files0[entries1[str0]] = mkfile(m1, md3, # links(entries1[str0], dirs0), content), files0[entries1[str]] = mkfile(m5, md0, m11 + 1, content1), dirs0[m0] = mkdir(m19 + 1, md4, # links(m0, dirs0), # subdirs(m0, dirs0), entries1), files0[m0] = mkfile(m12, md7, m13, content0), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, entries1[str] ∈ files0, str ∈ entries1, entries1[str0] ∈ files0, str0 ∈ entries1, m0 ∈ dirs0, # links(m0, dirs0) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files0, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries1 = ∅, m19 = # (entries1 -- str0)[str, entries1[str0]], mkdentry(str, entries1[str]) = mkdentry(str0, entries1[str0]), entries1[str] ∈ dirs0, entries1[str0] ∈ dirs0, m0 ≠ ROOT_INO ∧ links(m0, dirs0) = ∅

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 131:
⊦ ¬ 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, entries1 -- str0, entries1[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, entries1 -- str0, entries1[str0]