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Proof step 45 from proof writepage-prepost (previous step)

# σ = ∞, # buf0 = n4 + n0 + n1, ((m0 =>mkfile(n2, md, m1, σ) * F))(files), afs-cons(dirs, files), m0 ∈ files, n0 + n1 ≤ VFS_PAGE_SIZE, m + n1 ≤ # buf1, n0 + n1 ≤ # buf0, n0 ≤ VFS_PAGE_SIZE ⊦ ((σ to n3 * VFS_PAGE_SIZE) + ((σ from n3 * VFS_PAGE_SIZE) to n0)) + ((⌜buf1⌝ from m) to n1) + (((σ from n3 * VFS_PAGE_SIZE) to VFS_PAGE_SIZE) from n0 + n1) + (σ from VFS_PAGE_SIZE * n3 + VFS_PAGE_SIZE) = (σ to n3 * VFS_PAGE_SIZE + n0) + ((⌜buf1⌝ from m) to n1) + (σ from n3 * VFS_PAGE_SIZE + n0 + n1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ ¬ # σ0 < ⌜n + n1⌝ → (σ0 to n) + ((σ0 from n) to n1) = σ0 to n + n1
Proof step 47:
# buf0 = n4 + n0 + n1, # σ = ∞, ((m0 =>mkfile(n2, md, m1, σ) * F))(files), afs-cons(dirs, files), m0 ∈ files, n0 ≤ VFS_PAGE_SIZE, n0 + n1 ≤ # buf0, m + n1 ≤ # buf1, n0 + n1 ≤ VFS_PAGE_SIZE ⊦ (σ to n3 * VFS_PAGE_SIZE + n0) + ((⌜buf1⌝ from m) to n1) + (((σ from n3 * VFS_PAGE_SIZE) to VFS_PAGE_SIZE) from n0 + n1) + (σ from VFS_PAGE_SIZE * n3 + VFS_PAGE_SIZE) = (σ to n3 * VFS_PAGE_SIZE + n0) + ((⌜buf1⌝ from m) to n1) + (σ from n3 * VFS_PAGE_SIZE + n0 + n1), # σ < ⌜n3 * VFS_PAGE_SIZE + n0⌝

Rule argument

spec: stream, inst: buffer-inf name: from-append-tofrom seq: ⊦ ¬ # σ0 < ⌜n + n1⌝ → (σ0 to n) + ((σ0 from n) to n1) = σ0 to n + n1 subst: [n1, n, σ0] → [n0, n3 * VFS_PAGE_SIZE, σ] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜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) Used terms: n0, n3 * VFS_PAGE_SIZE, σ