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Proof step 9 from proof subdirs-put (previous step)

dirs[p_ino] = mkdir(m, md, n0, n1, entries), p_ino ∈ dirs ⊦ str ∈ (ino ∈ dirs ⊃ subdirs(p_ino, dirs) ++ str;(str ∈ entries ∧ entries[str] ∈ dirs ⊃ subdirs(p_ino, dirs) -- str;subdirs(p_ino, dirs))) ↔ str ∈ (subdirs(p_ino, dirs) \ (str ∈ entries ∧ entries[str] ∈ dirs ⊃ ∅ ++ str;∅)) ∪ (ino ∈ dirs ⊃ ∅ ++ str;∅), ino ∈ dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
dirs[p_ino] = mkdir(m, md, n0, n1, entries), p_ino ∈ dirs ⊦ str ∈ (str ∈ entries ∧ entries[str] ∈ dirs ⊃ subdirs(p_ino, dirs) -- str;subdirs(p_ino, dirs)) ↔ str ∈ subdirs(p_ino, dirs) \ (str ∈ entries ∧ entries[str] ∈ dirs ⊃ ∅ ++ str;∅), ino ∈ dirs

Rule argument

Additional information

Used simplifier rules: unc: ⊦ strings ∪ strings0 = strings0 ∪ strings; from specification set-union[dir-entries]%una: ⊦ (strings ∪ strings0) ∪ strings1 = strings ∪ strings0 ∪ strings1; from specification set-union[dir-entries]% ⊦ strings ∪ ∅ = strings; (s)