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Proof step 3 from proof overwrite-write-inode-dirty (previous step)

icache[ino] = mkientry(boolvar, inode), ino ∈ icache, ino ∈ files ⊦ ((files ⊕ pcache) ⊕ icache)[ino, (((files ⊕ pcache) ⊕ icache)[ino].meta:= md).size:= m] = (files ⊕ pcache) ⊕ icache[ino, mkientry(true, (inode.meta:= md).size:= m)], inode.directory

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 5:
icache[ino] = mkientry(boolvar, inode), ino ∈ files, ino ∈ icache ⊦ inode.directory, ∀ ino0. ( ino0 ∈ ((files ⊕ pcache) ⊕ icache)[ino, (((files ⊕ pcache) ⊕ icache)[ino].meta:= md).size:= m] ↔ ino0 ∈ (files ⊕ pcache) ⊕ icache[ino, mkientry(true, (inode.meta:= md).size:= m)]) ∧ ( ino0 ∈ ((files ⊕ pcache) ⊕ icache)[ino, (((files ⊕ pcache) ⊕ icache)[ino].meta:= md).size:= m] → ((files ⊕ pcache) ⊕ icache)[ino, (((files ⊕ pcache) ⊕ icache)[ino].meta:= md).size:= m][ino0] = ((files ⊕ pcache) ⊕ icache[ino, mkientry(true, (inode.meta:= md).size:= m)])[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(true, (inode.meta:= md).size:= m)], ((files ⊕ pcache) ⊕ icache )[ino, (((files ⊕ pcache) ⊕ icache)[ino].meta:= md).size:= m ]] flags: <all paths>

Additional information

Used terms: (files ⊕ pcache) ⊕ icache[ino, mkientry(true, (inode.meta:= md).size:= m)], ((files ⊕ pcache) ⊕ icache )[ino, (((files ⊕ pcache) ⊕ icache)[ino].meta:= md).size:= m ]