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Proof step 149 from proof afs-cons-rename-dir-negdentry (previous step)

# subdirs(m, dirs0) = m2 + 1, # entries2 = m4 + 1, dirs0[m] = mkdir(m4 + 1, md, # links(m, dirs0), m2 + 1, entries2), dirs0[entries2[str]] = mkdir(m3, md4, # links(entries2[str], dirs0), m5, entries1), files[m] = mkfile(m9, md2, m11, content), afs-cons(dirs0, files), str ∈ entries2, entries2[str] ∈ dirs0, ROOT_INO ∈ dirs0, m ∈ dirs0, # links(m, dirs0) ≤ 1, name ∈ entries2 → entries2[name] ∈ dirs0 ∨ entries2[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ m4 + 1 = # (entries2 -- str)[str0, entries2[str]], entries2 = ∅, str0 ∈ entries2, m ≠ ROOT_INO ∧ links(m, dirs0) = ∅

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

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