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Proof step 43 from proof fsync-inode-keep-abs (previous step)

dirs[m0] = mkdir(m1, md0, m2, m3, entries), icache[m] = mkientry(true, mkinode(m0, md, true, m2, m3, m1)), icache-cons(icache, dirs, files), m ∈ icache, m0 ∈ dirs ⊦ (files ⊕ pcache) ⊕ icache = (files ⊕ pcache) ⊕ icache[m, mkientry(false, mkinode(m0, md, true, m2, m3, m1))]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 45:
icache[m] = mkientry(true, mkinode(m0, md, true, m2, m3, m1)), dirs[m0] = mkdir(m1, md0, m2, m3, entries), icache-cons(icache, dirs, files), m0 ∈ dirs, m ∈ icache ⊦ ∀ ino. (ino ∈ (files ⊕ pcache) ⊕ icache ↔ ino ∈ (files ⊕ pcache) ⊕ icache[m, mkientry(false, mkinode(m0, md, true, m2, m3, m1))]) ∧ ( ino ∈ (files ⊕ pcache) ⊕ icache → ((files ⊕ pcache) ⊕ icache)[ino] = ((files ⊕ pcache) ⊕ icache[m, mkientry(false, mkinode(m0, md, true, m2, m3, m1))])[ino])

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 [m, mkientry(false, mkinode(m0, md, true, m2, m3, m1)) ], (files ⊕ pcache) ⊕ icache] flags: <all paths>

Additional information

Used terms: (files ⊕ pcache) ⊕ icache[m, mkientry(false, mkinode(m0, md, true, m2, m3, m1))], (files ⊕ pcache) ⊕ icache