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Proof step 93 from proof truncate-inv (previous step)

VFS_PAGE_SIZE = n1 + n0, files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), n0 < VFS_PAGE_SIZE, afs-cons (dirs0, files2 [ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0[m2, fill(content0[m2], zero, n0, n1)] upto (m2 * VFS_PAGE_SIZE + n0)) ]), afs-cons(dirs0, files2), files2 ~ files2 [ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0[m2, fill(content0[m2], zero, n0, n1)] upto (m2 * VFS_PAGE_SIZE + n0)) ], m2 ∈ content0, ino4 ∈ files2 ⊦ ( (content0.stream to m2 * VFS_PAGE_SIZE) + ((content0.stream from m2 * VFS_PAGE_SIZE) to n0) + (σ⊥ to n1) + (content0.stream from VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE) to VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE) + σ⊥ from m2 * VFS_PAGE_SIZE + n0 = σ⊥, n0 = 0, n1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 94:
⊦ m ≤ n ∧ ¬ # σ1 < ⌜n⌝ → (σ1 to n) + σ2 from m = ((σ1 from m) to (n - m)) + σ2
Proof step 95:
VFS_PAGE_SIZE = n1 + n0, files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), n0 < VFS_PAGE_SIZE, afs-cons(dirs0, files2), afs-cons (dirs0, files2 [ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0[m2, fill(content0[m2], zero, n0, n1)] upto (m2 * VFS_PAGE_SIZE + n0)) ]), files2 ~ files2 [ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0[m2, fill(content0[m2], zero, n0, n1)] upto (m2 * VFS_PAGE_SIZE + n0)) ], ino4 ∈ files2, m2 ∈ content0, m2 * VFS_PAGE_SIZE + n0 ≤ VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE ⊦ ( ( (content0.stream to m2 * VFS_PAGE_SIZE) + ((content0.stream from m2 * VFS_PAGE_SIZE) to n0) + (σ⊥ to n1) + (content0.stream from VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE) from m2 * VFS_PAGE_SIZE + n0) to (VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE - m2 * VFS_PAGE_SIZE + n0)) + σ⊥ = σ⊥, n0 = 0, n1 = 0, # ( (content0.stream to m2 * VFS_PAGE_SIZE) + ((content0.stream from m2 * VFS_PAGE_SIZE) to n0) + (σ⊥ to n1) + (content0.stream from VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE)) < ⌜VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE⌝

Rule argument

spec: stream, inst: buffer-inf name: swap-to-from-app seq: ⊦ m ≤ n ∧ ¬ # σ1 < ⌜n⌝ → (σ1 to n) + σ2 from m = ((σ1 from m) to (n - m)) + σ2 subst: [m, σ2, n, σ1] → [m2 * VFS_PAGE_SIZE + n0, σ⊥, VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE, (content0.stream to m2 * VFS_PAGE_SIZE) + ((content0.stream from m2 * VFS_PAGE_SIZE) to n0) + (σ⊥ to n1) + (content0.stream from VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE)] flags: <all paths>

Additional information

Used simplifier rules: mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (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% ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # content.stream = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ #(content.stream from n) = ∞; (s)% ⊦ # σ⊥ = ∞; (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%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) Used terms: m2 * VFS_PAGE_SIZE + n0, σ⊥, VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE, (content0.stream to m2 * VFS_PAGE_SIZE) + ((content0.stream from m2 * VFS_PAGE_SIZE) to n0) + (σ⊥ to n1) + (content0.stream from VFS_PAGE_SIZE * m2 + VFS_PAGE_SIZE)