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Theorem subdirs-put

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

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