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Proof step 41 from proof truncate-empty (previous step)

VFS_PAGE_SIZE = n0 + n1, files1[ino0] = mkfile(m, md0, # links(ino0, dirs), content0), n0 ≠ 0, n1 ≠ 0, n1 < VFS_PAGE_SIZE, afs-cons(dirs, files1), ino0 ∈ files1, m0 ∈ content0 ⊦ [asynced := ?] ((content0[m0, fill(content0[m0], zero, n1, n0)] upto (m0 * VFS_PAGE_SIZE + n1)).stream from m0 * VFS_PAGE_SIZE + n1 = σ⊥)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ n = n0 * VFS_PAGE_SIZE + n1 ∧ n1 ≠ 0 ∧ n1 < VFS_PAGE_SIZE → (content upto n).stream = (content.stream to (n0 + 1) * VFS_PAGE_SIZE) + σ⊥
Proof step 43:
m0 * VFS_PAGE_SIZE + n1 = m0 * VFS_PAGE_SIZE + n1, VFS_PAGE_SIZE = n0 + n1, files1[ino0] = mkfile(m, md0, # links(ino0, dirs), content0), n1 < VFS_PAGE_SIZE, afs-cons(dirs, files1), m0 ∈ content0, ino0 ∈ files1 ⊦ [asynced := ?] ((content0[m0, fill(content0[m0], zero, n1, n0)].stream to (m0 + 1) * VFS_PAGE_SIZE) + σ⊥ from m0 * VFS_PAGE_SIZE + n1 = σ⊥), n0 = 0, n1 = 0

Rule argument

spec: afs-stream name: content-trunc seq: ⊦ n = n0 * VFS_PAGE_SIZE + n1 ∧ n1 ≠ 0 ∧ n1 < VFS_PAGE_SIZE → (content upto n).stream = (content.stream to (n0 + 1) * VFS_PAGE_SIZE) + σ⊥ subst: [n0, n1, n, content] → [m0, n1, m0 * VFS_PAGE_SIZE + n1, content0[m0, fill(content0[m0], zero, n1, n0)]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ afs-cons(dirs, files) ∧ ino ∈ files ∧ files[ino] = mkfile(m, md, n0, content) ∧ n ∈ content → # content[n] = VFS_PAGE_SIZE; (s)% ⊦ m * (n + 1) = m * n + m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)% ⊦ # content.stream = ∞; (s)% ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s)% ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used terms: m0, n1, m0 * VFS_PAGE_SIZE + n1, content0[m0, fill(content0[m0], zero, n1, n0)]