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Proof step 17 from proof synced-dirs-add-update-clean-inode (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 18:
icache[ino] = mkientry(false, inode0), pcache[datakey(ino0, n)] = mkpentry(boolvar, buf), ino ∈ icache, ino0 ∈ dirs, ∀ ino0. ino0 ∈ dirs → (ino0 ∈ icache → ¬ icache[ino0].dirty) ∧ (∀ n. datakey(ino0, n) ∈ pcache → ¬ pcache[datakey(ino0, n)].dirty) ⊦ (ino0 ∈ icache[ino, mkientry(false, inode)] → ¬ icache[ino, mkientry(false, inode)][ino0].dirty) ∧ (datakey(ino0, n) ∈ pcache → ¬ boolvar)

Rule argument

spec: pcache-entry name: elim seq: ⊦ (boolvar ↔ pe.dirty) ∧ buf = pe.page ↔ pe = mkpentry(boolvar, buf) subst: [pe, boolvar, buf] → [pcache[datakey(ino0, n)], boolvar, buf]

Additional information

Used simplifier rules: ⊦ (boolvar ↔ pe.dirty) ∧ buf = pe.page ↔ pe = mkpentry(boolvar, buf); (elim) Used terms: pcache[datakey(ino0, n)], boolvar, buf