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Proof step 5 from proof totality (previous step)

key0.ino ≤ ino ⊦ inodekey(ino) = key0, key0 < inodekey(ino), ¬ key0.inode? ∧ ino = key0.ino

User applied rule constructor cut

And closed the goal!

Rule argument

var: key0

Additional information

Used simplifier rules: key freely generated by inodekey, datakey, dentrykey;;% ⊦ inodekey(m) ≠ dentrykey(m0, str); (s)% ⊦ dentrykey(m, str).ino = m; (s)%⊦ dentrykey(ino, str) < key ↔ ino < key.ino ∨ key.dentry? ∧ ino = key.ino ∧ str < key.name ∨ key.data? ∧ ino = key.ino; (ws)% ⊦ inodekey(m).ino = m; (s)% ⊦ ¬ inodekey(m).dentry?; (s)% ⊦ ¬ inodekey(m).data?; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)% ⊦ ¬ dentrykey(m, str).inode?; (s)% ⊦ inodekey(m) ≠ datakey(m0, m1); (s)% ⊦ datakey(m, m0).ino = m; (s)%⊦ datakey(ino, n) < key ↔ ino < key.ino ∨ key.data? ∧ ino = key.ino ∧ n < key.part; (ws)% ⊦ ¬ datakey(m, m0).inode?; (s)% ⊦ inodekey(m) = inodekey(m0) ↔ m = m0; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ inodekey(ino) < key ↔ ino < key.ino ∨ ¬ key.inode? ∧ ino = key.ino; (ws)