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Proof step 6 from proof overwrite-write-inode-clean-dir (previous step)

icache[ino] = mkientry(boolvar, mkinode(m, md0, true, m0, m1, m2)), ino ∈ icache, ino ∈ dirs ⊦ (files ⊕ pcache) ⊕ icache = (files ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m, md, true, m0, m1, m2))]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 8:
icache[ino] = mkientry(boolvar, mkinode(m, md0, true, m0, m1, m2)), ino ∈ dirs, ino ∈ icache ⊦ ∀ ino0. (ino0 ∈ (files ⊕ pcache) ⊕ icache ↔ ino0 ∈ (files ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m, md, true, m0, m1, m2))]) ∧ ( ino0 ∈ (files ⊕ pcache) ⊕ icache → ((files ⊕ pcache) ⊕ icache)[ino0] = ((files ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m, md, true, m0, m1, m2))])[ino0])

Rule argument

spec: heap, inst: file-heap name: ext seq: ⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino])) subst: [files0, files] → [ (files ⊕ pcache) ⊕ icache [ino, mkientry(false, mkinode(m, md, true, m0, m1, m2)) ], (files ⊕ pcache) ⊕ icache] flags: <all paths>

Additional information

Used terms: (files ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m, md, true, m0, m1, m2))], (files ⊕ pcache) ⊕ icache