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Proof step 9 from proof overwrite-write-inode-clean-file (previous step)

icache[ino] = mkientry(boolvar, mkinode(m0, md0, false, m1, m2, m3)), files[ino] = mkfile(m4, md1, m5, content), ino ∈ icache, ino ∈ files ⊦ ((files ⊕ pcache) ⊕ icache)[ino, mkfile(m, md, m1, content ⊕ mkcontent(ino, pcache))] = (files[ino, mkfile(m, md, m5, content)] ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m0, md, false, m1, m2, m))]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 11:
files[ino] = mkfile(m4, md1, m5, content), icache[ino] = mkientry(boolvar, mkinode(m0, md0, false, m1, m2, m3)), ino ∈ files, ino ∈ icache ⊦ ∀ ino0. ( ino0 ∈ ((files ⊕ pcache) ⊕ icache)[ino, mkfile(m, md, m1, content ⊕ mkcontent(ino, pcache))] ↔ ino0 ∈ (files[ino, mkfile(m, md, m5, content)] ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m0, md, false, m1, m2, m))]) ∧ ( ino0 ∈ ((files ⊕ pcache) ⊕ icache)[ino, mkfile(m, md, m1, content ⊕ mkcontent(ino, pcache))] → ((files ⊕ pcache) ⊕ icache)[ino, mkfile(m, md, m1, content ⊕ mkcontent(ino, pcache))][ino0] = ((files[ino, mkfile(m, md, m5, content)] ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m0, md, false, m1, m2, 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[ino, mkfile(m, md, m5, content)] ⊕ pcache) ⊕ icache [ino, mkientry(false, mkinode(m0, md, false, m1, m2, m)) ], ((files ⊕ pcache) ⊕ icache )[ino, mkfile(m, md, m1, content ⊕ mkcontent(ino, pcache)) ]] flags: <all paths>

Additional information

Used terms: (files[ino, mkfile(m, md, m5, content)] ⊕ pcache) ⊕ icache[ino, mkientry(false, mkinode(m0, md, false, m1, m2, m))], ((files ⊕ pcache) ⊕ icache )[ino, mkfile(m, md, m1, content ⊕ mkcontent(ino, pcache)) ]