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Proof step 17 from proof write-inv-rec (previous step)

n0 + n = m0 * VFS_PAGE_SIZE + n2, files1 -- ino = files0 -- ino, content0.stream = (content.stream to n0) + (⌜buf⌝ to n) + (content.stream from n0 + n), files1[ino] = mkfile(m2, md0, # links(ino, dirs), content0), files0[ino] = mkfile(m2, md0, m4, content), n2 < VFS_PAGE_SIZE, afs-cons(dirs, files1), ino ∈ files1, ino ∈ files0, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1, m + n2 ≤ VFS_PAGE_SIZE, n + m ≤ # buf ⊦ (content0.stream to m0 * VFS_PAGE_SIZE) + ⌜copy(buf, n, getpage(m0, content0), n2, m)⌝ + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) = (content.stream to n0) + (⌜buf⌝ to n + m) + (content.stream from n0 + n + m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ n0 + m ≤ # buf0 ∧ n1 + m ≤ # buf1 → ⌜copy(buf0, n0, buf1, n1, m)⌝ = (⌜buf1⌝ to n1) + ((⌜buf0⌝ from n0) to m) + ((⌜buf1⌝ from n1 + m) to (# buf1 - n1 + m))
Proof step 19:
n0 + n = m0 * VFS_PAGE_SIZE + n2, files1 -- ino = files0 -- ino, content0.stream = (content.stream to n0) + (⌜buf⌝ to n) + (content.stream from n0 + n), files0[ino] = mkfile(m2, md0, m4, content), files1[ino] = mkfile(m2, md0, # links(ino, dirs), content0), n2 < VFS_PAGE_SIZE, afs-cons(dirs, files1), ino ∈ files0, ino ∈ files1, m + n2 ≤ VFS_PAGE_SIZE, n0 + n ≤ n1, n1 ≤ # buf + n0, n ≤ # buf, n + m ≤ # buf, n2 + m ≤ # getpage(m0, content0) ⊦ (content0.stream to m0 * VFS_PAGE_SIZE) + ((⌜getpage(m0, content0)⌝ to n2) + ((⌜buf⌝ from n) to m) + ((⌜getpage(m0, content0)⌝ from n2 + m) to (# getpage(m0, content0) - n2 + m))) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) = (content.stream to n0) + (⌜buf⌝ to n + m) + (content.stream from n0 + n + m)

Rule argument

spec: afs-stream name: copy-stream-append seq: ⊦ n0 + m ≤ # buf0 ∧ n1 + m ≤ # buf1 → ⌜copy(buf0, n0, buf1, n1, m)⌝ = (⌜buf1⌝ to n1) + ((⌜buf0⌝ from n0) to m) + ((⌜buf1⌝ from n1 + m) to (# buf1 - n1 + m)) subst: [m, n1, buf1, n0, buf0] → [m, n2, getpage(m0, content0), n, buf] flags: <all paths>

Additional information

Used simplifier rules: ⊦ afs-cons(dirs, files) ∧ ino ∈ files ∧ files[ino] = mkfile(m, md, m0, content) → # getpage(n, content) = VFS_PAGE_SIZE; (s)%c: ⊦ m + n = n + m; from specification nat-basic Used terms: m, n2, getpage(m0, content0), n, buf