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Proof step 47 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.stream to m0 * VFS_PAGE_SIZE) + ⌜fill(content0[m0], zero, n1, n0)⌝ + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) to VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) + σ⊥ from m0 * VFS_PAGE_SIZE + n1 = σ⊥)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ n + m = # buf → ⌜fill(buf, zero, n, m)⌝ = (⌜buf⌝ to n) + (σ⊥ to m)
Proof step 49:
n1 + n0 = # content0[m0], 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.stream to m0 * VFS_PAGE_SIZE) + ((⌜content0[m0]⌝ to n1) + (σ⊥ to n0)) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) to VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) + σ⊥ from m0 * VFS_PAGE_SIZE + n1 = σ⊥), n0 = 0, n1 = 0

Rule argument

spec: afs-stream name: fill-stream-append seq: ⊦ n + m = # buf → ⌜fill(buf, zero, n, m)⌝ = (⌜buf⌝ to n) + (σ⊥ to m) subst: [m, n, buf] → [n0, n1, content0[m0]] 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)%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ # σ⊥ = ∞; (s)% ⊦ ⌜m⌝ < ∞; (s)% ⊦ finp(⌜buf⌝ ); (ss)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic% ⊦ m < n → ¬ n < m; (s) Used terms: n0, n1, content0[m0]