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Proof step 78 from proof fsync-pages-others-unchanged (previous step)

# buf0 = VFS_PAGE_SIZE, files1[ino0] = mkfile(m5, md1, m6, content0), files1[m] = mkfile(m0, md, m1, content), pcache[datakey(m, n)] = mkpentry(true, buf0), files0[ino0] = mkfile(m5, md1, m6, content1), (content0 ⊕ mkcontent(ino0, pcache))[n0] = (content1 ⊕ mkcontent(ino0, pcache0))[n0], n * VFS_PAGE_SIZE < m2, m ∈ files1, datakey(m, n) ∈ pcache, ino0 ∈ files0, ino0 ∈ files1, n0 ∈ content0 ⊕ mkcontent(ino0, pcache), n0 ∈ content1 ⊕ mkcontent(ino0, pcache0), n0 ∈ content0, ∀ n. (n ∈ content0 ⊕ mkcontent(ino0, pcache) ↔ n ∈ content1 ⊕ mkcontent(ino0, pcache0)) ∧ (n ∈ content0 ⊕ mkcontent(ino0, pcache) → (content0 ⊕ mkcontent(ino0, pcache))[n] = (content1 ⊕ mkcontent(ino0, pcache0))[n]), ∀ ino. ino ≠ m ∧ ino ∈ files0 → mkfile(files1[ino].size, files1[ino].meta, files1[ino].nlink, files1[ino].content ⊕ mkcontent(ino, pcache)) = mkfile(files0[ino].size, files0[ino].meta, files0[ino].nlink, files0[ino].content ⊕ mkcontent(ino, pcache0)) ∧ ino ∈ files1 ⊦ content0[n0] = (content1 ⊕ mkcontent(ino0, pcache0))[n0], datakey(ino0, n0) = datakey(m, n), ino0 = m, datakey(ino0, n0) ∈ pcache

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 79:
⊦ ¬ n ∈ content1 → (content0 ⊕ content1)[n] = content0[n]
Proof step 80:
# buf0 = VFS_PAGE_SIZE, (content0 ⊕ mkcontent(ino0, pcache))[n0] = (content1 ⊕ mkcontent(ino0, pcache0))[n0], files0[ino0] = mkfile(m5, md1, m6, content1), pcache[datakey(m, n)] = mkpentry(true, buf0), files1[m] = mkfile(m0, md, m1, content), files1[ino0] = mkfile(m5, md1, m6, content0), n * VFS_PAGE_SIZE < m2, n0 ∈ mkcontent(ino0, pcache0), n0 ∈ content0, n0 ∈ content1 ⊕ mkcontent(ino0, pcache0), n0 ∈ content0 ⊕ mkcontent(ino0, pcache), ino0 ∈ files1, ino0 ∈ files0, datakey(m, n) ∈ pcache, m ∈ files1, ∀ ino. ino ≠ m ∧ ino ∈ files0 → mkfile(files1[ino].size, files1[ino].meta, files1[ino].nlink, files1[ino].content ⊕ mkcontent(ino, pcache)) = mkfile(files0[ino].size, files0[ino].meta, files0[ino].nlink, files0[ino].content ⊕ mkcontent(ino, pcache0)) ∧ ino ∈ files1, ∀ n. (n ∈ content0 ⊕ mkcontent(ino0, pcache) ↔ n ∈ content1 ⊕ mkcontent(ino0, pcache0)) ∧ (n ∈ content0 ⊕ mkcontent(ino0, pcache) → (content0 ⊕ mkcontent(ino0, pcache))[n] = (content1 ⊕ mkcontent(ino0, pcache0))[n]) ⊦ content0[n0] = (content1 ⊕ mkcontent(ino0, pcache0))[n0], datakey(ino0, n0) = datakey(m, n), ino0 = m, datakey(ino0, n0) ∈ pcache
Proof step 85:
# buf0 = VFS_PAGE_SIZE, (content0 ⊕ mkcontent(ino0, pcache))[n0] = (content1 ⊕ mkcontent(ino0, pcache0))[n0], files0[ino0] = mkfile(m5, md1, m6, content1), pcache[datakey(m, n)] = mkpentry(true, buf0), files1[m] = mkfile(m0, md, m1, content), files1[ino0] = mkfile(m5, md1, m6, content0), n * VFS_PAGE_SIZE < m2, n0 ∈ content0, n0 ∈ content1 ⊕ mkcontent(ino0, pcache0), n0 ∈ content0 ⊕ mkcontent(ino0, pcache), ino0 ∈ files1, ino0 ∈ files0, datakey(m, n) ∈ pcache, m ∈ files1, ∀ ino. ino ≠ m ∧ ino ∈ files0 → mkfile(files1[ino].size, files1[ino].meta, files1[ino].nlink, files1[ino].content ⊕ mkcontent(ino, pcache)) = mkfile(files0[ino].size, files0[ino].meta, files0[ino].nlink, files0[ino].content ⊕ mkcontent(ino, pcache0)) ∧ ino ∈ files1, ∀ n. (n ∈ content0 ⊕ mkcontent(ino0, pcache) ↔ n ∈ content1 ⊕ mkcontent(ino0, pcache0)) ∧ (n ∈ content0 ⊕ mkcontent(ino0, pcache) → (content0 ⊕ mkcontent(ino0, pcache))[n] = (content1 ⊕ mkcontent(ino0, pcache0))[n]) ⊦ content0[n0] = content1[n0], datakey(ino0, n0) = datakey(m, n), ino0 = m, datakey(ino0, n0) ∈ pcache, n0 ∈ mkcontent(ino0, pcache0)

Rule argument

spec: cache-base name: at-overwrite-content-old seq: ⊦ ¬ n ∈ content1 → (content0 ⊕ content1)[n] = content0[n] subst: [n, content1, content0] → [n0, mkcontent(ino0, pcache0), content1] flags: List([[1,0,0,1,1]])

Additional information

Used terms: n0, mkcontent(ino0, pcache0), content1