[Imprint][Data Privacy Statement][E-Mail]

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Proof step 62 from proof truncate-greater (previous step)

VFS_PAGE_SIZE = n0 + n2, files0[m0] = mkfile(m, md0, # links(m0, dirs), content0), n2 < VFS_PAGE_SIZE, afs-cons(dirs, files0), m1 ∈ content0, m0 ∈ files0 ⊦ [asynced := ?] ( ( ((content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n0)) + (content0.stream from VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE) to VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE) + σ⊥ to n1 + m1 * VFS_PAGE_SIZE + n2 = (content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n1)), n0 = 0, n1 = 0, n2 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 63:
⊦ ¬ # σ1 < ⌜n⌝ → σ1 + σ2 to n = σ1 to n
Proof step 64:
VFS_PAGE_SIZE = n0 + n2, files0[m0] = mkfile(m, md0, # links(m0, dirs), content0), n2 < VFS_PAGE_SIZE, afs-cons(dirs, files0), m0 ∈ files0, m1 ∈ content0 ⊦ [asynced := ?] ( ((content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n0) to VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE) + σ⊥ to n1 + m1 * VFS_PAGE_SIZE + n2 = (content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n1)), n0 = 0, n1 = 0, n2 = 0, #((content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n0)) < ⌜VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE⌝

Rule argument

spec: stream, inst: buffer-inf name: append-to-ls seq: ⊦ ¬ # σ1 < ⌜n⌝ → σ1 + σ2 to n = σ1 to n subst: [n, σ2, σ1] → [VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE, content0.stream from VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE, (content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # 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)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ # σ⊥ = ∞; (s)%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s)% ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (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% ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s)% ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)% ⊦ ⌜m⌝ < ∞; (s)% ⊦ #(content.stream from n) = ∞; (s)% ⊦ ⌜m⌝ + ∞ = ∞; (s)% ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s)% ⊦ ⌜n⌝ + n∞1 = ⌜n⌝ + n∞2 ↔ n∞1 = n∞2; (s)% ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s)%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: VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE, content0.stream from VFS_PAGE_SIZE * m1 + VFS_PAGE_SIZE, (content0.stream to m1 * VFS_PAGE_SIZE + n2) + (σ⊥ to n0)