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Proof step 66 from proof truncate-empty (previous step)

VFS_PAGE_SIZE = n2 + n1, files1[ino0] = mkfile(m, md0, # links(ino0, dirs), content0), n1 ≠ 0, n2 ≠ 0, n1 < VFS_PAGE_SIZE, afs-cons(dirs, files1), m0 ∈ content0, ino0 ∈ files1 ⊦ [asynced := ?] ( ( ( (content0.stream to m0 * VFS_PAGE_SIZE + n1) + (σ⊥ to n2) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) to (m0 * VFS_PAGE_SIZE + n1) + n2) from m0 * VFS_PAGE_SIZE + n1) + σ⊥ = σ⊥)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 67:
⊦ ¬ # σ1 < ⌜n⌝ ∧ ¬ # σ2 < ⌜n1⌝ → (σ1 to n) + σ2 to n1 + n = (σ1 to n) + (σ2 to n1)
Proof step 68:
VFS_PAGE_SIZE = n2 + n1, files1[ino0] = mkfile(m, md0, # links(ino0, dirs), content0), n1 < VFS_PAGE_SIZE, afs-cons(dirs, files1), ino0 ∈ files1, m0 ∈ content0 ⊦ [asynced := ?] ( ( (content0.stream to m0 * VFS_PAGE_SIZE + n1) + ((σ⊥ to n2) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE) to n2) from m0 * VFS_PAGE_SIZE + n1) + σ⊥ = σ⊥), n1 = 0, n2 = 0, #((σ⊥ to n2) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE)) < ⌜n2⌝ , # content0.stream < ⌜m0 * VFS_PAGE_SIZE + n1⌝

Rule argument

spec: stream, inst: buffer-inf name: to-append-add seq: ⊦ ¬ # σ1 < ⌜n⌝ ∧ ¬ # σ2 < ⌜n1⌝ → (σ1 to n) + σ2 to n1 + n = (σ1 to n) + (σ2 to n1) subst: [n1, σ2, n, σ1] → [n2, (σ⊥ to n2) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE), m0 * VFS_PAGE_SIZE + n1, content0.stream] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # content.stream = ∞; (s)% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)% ⊦ ¬ ∞ < n∞; (s)% ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # σ⊥ = ∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ #(content.stream from 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% ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s)% ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)% ⊦ ⌜m⌝ < ∞; (s)%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n2, (σ⊥ to n2) + (content0.stream from VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE), m0 * VFS_PAGE_SIZE + n1, content0.stream