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Proof step 68 from proof write-inv-rec (previous step)

n0 + n = m0 * VFS_PAGE_SIZE + n2, VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE = n3 + n0 + n, files1 -- ino = files0 -- ino, VFS_PAGE_SIZE = n4 + n2 + m, 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, m0 * VFS_PAGE_SIZE ≤ n3 + n0 + n, m0 * VFS_PAGE_SIZE ≤ # buf + n0, m + n2 ≤ VFS_PAGE_SIZE, n0 + n ≤ n1, n1 ≤ # buf + n0, n ≤ # buf, n + m ≤ # buf, m0 * VFS_PAGE_SIZE ≤ n + n0, m0 * VFS_PAGE_SIZE ≤ n1, n2 + m ≤ VFS_PAGE_SIZE ⊦ n4 + m = 0, ((content.stream to n0) + (⌜buf⌝ to n) to m0 * VFS_PAGE_SIZE) + (((content.stream to n0) + (⌜buf⌝ to n) to n0 + n) from m0 * VFS_PAGE_SIZE) + ((⌜buf⌝ from n) to m) + ((content.stream from n0 + n from m) to n4) + (content.stream from n0 + n from n3) = (content.stream to n0) + (⌜buf⌝ to n + m) + (content.stream from n0 + n + m), #(((content.stream to n0) + (⌜buf⌝ to n)) + (content.stream from n0 + n) from m0 * VFS_PAGE_SIZE) < ⌜VFS_PAGE_SIZE⌝ , #(content.stream from n0 + n from m) < ⌜n4⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 69:
⊦ ¬ # σ0 < ⌜m + n⌝ → σ0 from m from n = σ0 from m + n
Proof step 70:
VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE = n3 + n0 + n, n0 + n = m0 * VFS_PAGE_SIZE + n2, files1 -- ino = files0 -- ino, VFS_PAGE_SIZE = n4 + n2 + m, 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, n2 + m ≤ VFS_PAGE_SIZE, m0 * VFS_PAGE_SIZE ≤ n1, m0 * VFS_PAGE_SIZE ≤ n + n0, n + m ≤ # buf, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1, m + n2 ≤ VFS_PAGE_SIZE, m0 * VFS_PAGE_SIZE ≤ # buf + n0, m0 * VFS_PAGE_SIZE ≤ n3 + n0 + n ⊦ ((content.stream to n0) + (⌜buf⌝ to n) to m0 * VFS_PAGE_SIZE) + (((content.stream to n0) + (⌜buf⌝ to n) to n0 + n) from m0 * VFS_PAGE_SIZE) + ((⌜buf⌝ from n) to m) + ((content.stream from (n0 + n) + m) to n4) + (content.stream from n0 + n from n3) = (content.stream to n0) + (⌜buf⌝ to n + m) + (content.stream from n0 + n + m), n4 + m = 0, #(content.stream from (n0 + n) + m) < ⌜n4⌝ , #(((content.stream to n0) + (⌜buf⌝ to n)) + (content.stream from n0 + n) from m0 * VFS_PAGE_SIZE) < ⌜VFS_PAGE_SIZE⌝ , # content.stream < ⌜(n0 + n) + m⌝

Rule argument

spec: stream, inst: buffer-inf name: from-from seq: ⊦ ¬ # σ0 < ⌜m + n⌝ → σ0 from m from n = σ0 from m + n subst: [n, m, σ0] → [m, n0 + n, content.stream] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # content.stream = ∞; (s)% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ ∞ < n∞; (s) Used terms: m, n0 + n, content.stream