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Proof step 4 from proof synced-dirs-add-update-clean-inode (previous step)

icache[ino] = mkientry(false, inode0), ino ∈ icache ⊦ (∀ ino0. ino0 ∈ dirs → (ino0 ∈ icache[ino, mkientry(false, inode)] → ¬ icache[ino, mkientry(false, inode)][ino0].dirty) ∧ (∀ n. datakey(ino0, n) ∈ pcache → ¬ pcache[datakey(ino0, n)].dirty)) ↔ (∀ ino0. ino0 ∈ dirs → (ino0 ∈ icache → ¬ icache[ino0].dirty) ∧ (∀ n. datakey(ino0, n) ∈ pcache → ¬ pcache[datakey(ino0, n)].dirty))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

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